# Multiple strain-induced topological phase transitions in 1 D polyacene polymers: A first-principles study

M. Mosaferi, M. Aktas, D. Romanin, A. Chin

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M. Mosaferi, M. Aktas, D. Romanin, A. Chin. Multiple strain-induced topological phase transitions in 1 D polyacene polymers: A first-principles study. Physical Review Materials, 2025, 9 (4), pp.044204. 10.1103/PhysRevMaterials.9.044204. hal-05063730

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Submitted on 13 May 2025

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# Multiple Strain-Induced Topological Phase Transitions in 1D Polyacene Polymers: A First-Principles Study

M. Mosaferi¹,* M. Aktas²,† D. Romanin²,‡ and A. W. Chin¹§

¹Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252 Paris, France and

²Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

(Dated: May 13, 2025)

Applying both compressive and tensile strain, we explore via first-principles calculations how applied strain affects the band gap and charge density distribution within a class of recently synthesized topological polyacene polymers. Our findings reveal a strikingly non-monotonic behavior of the semiconducting band gap, characterized by several closures and reopenings at critical strain points, which can be rationalized by the presence of multiple topological phase transitions. Additionally, we also observe transitions between aromatic and quinonoid configurations, revealing qualitative structural changes as strain continuously tunes these systems across the rich landscape of trivial to non-trivial topological transitions, which includes at least two quasi-metallic phases. This highlights the pivotal role of strain in tuning the electronic properties of novel conjugated polymers, with promising implications for leveraging the flexible, non-toxic properties of organic materials with topological electronic effects for advanced electronic and optoelectronic devices.

## I. INTRODUCTION

Organic optoelectronic materials have garnered immense interest due to their versatility, light-harvesting efficiency [1–4], and chemical tunability [3, 4]. Their ability to combine lightweight, flexible, and environmentally friendly fabrication techniques has positioned them as key players in next-generation technologies [5, 6], such as photovoltaics, sensors, solid-state lighting [7–9], and organic solar cells [10, 11]. A particularly pressing area of research in this field is the development of low-bandgap organic polymers, essential for enhanced optical absorption and efficient charge transport.

In parallel, the study of topological materials has revolutionized condensed matter physics, providing deep insights into novel electronic states and robust properties. These materials, including topological insulators and superconductors, are characterized by conducting surface states that are protected by symmetries such as time-reversal or crystalline symmetries. These states are inherently linked to the topological nature of the band structure: when a material with a non-trivial topology interfaces with a trivial insulator (e.g., air), the bandgap must close and reopen at the boundary to connect the two phases, ensuring the presence of robust conducting states [12–15]. This fundamental result underscores the protection of such surface states against impurities and disorder. Moreover, if internal parameters such as strain or molecular design are tuned to change the material’s topology, the transition must pass through a gapless critical point, offering opportunities to explore unique transport phenomena. Organic polymers,

particularly those synthesized with atomistic precision through ‘on-surface’ methods [16, 17], are highly tunable platforms for achieving such transitions. This opens the possibility of developing robustly low- or zero-bandgap organic materials with exciting applications in quantum technologies and advanced optoelectronic devices.

Among these systems, polyacene polymers stand out as promising candidates for topological studies. Their π-conjugated structures and sensitivity to external perturbations, such as acene size and strain, provide a unique platform for investigating topological transitions [18–21]. Previous experimental efforts, including studies on single-bridge π-conjugated polymers, have demonstrated low-bandgap characteristics associated with topological transitions [22]. However, these transitions have so far been discretely tuned, often resulting in highly trivial or strongly non-trivial structures with limited practical applications. A key question that remains unanswered is how the electronic structure evolves close to the transition under continuous tuning parameters such as strain.

In this context, the ability to achieve ultra-low bandgaps through topological transitions represents an exciting opportunity for organic materials. Our study explores this potential by focusing on a double-bridged [n = 6] acene polymer (where n refers to the number of aromatic rings in each monomer). This system, positioned close to the critical point for topological phase transitions, bridges the gap between previously studied [n = 5] and [n = 7] acene polymers [19], exhibiting a low band gap at equilibrium (GW band gap = 0.29 eV). The introduction of a second bridge significantly alters the hopping parameters, reducing t₁ (intra-cell hopping) and increasing t₂ (inter-cell hopping). In the Su-Schrieffer-Heeger (SSH) model, the relative magnitudes of t₁ and t₂ govern the transition between topologically trivial and non-trivial phases. By decreasing t₁ and enhancing

<footnote>* mosaferi@insp.jussieu.fr</footnote>

<footnote>† melih.aktas@etu-upsaclay.fr</footnote>

<footnote>‡ davide.romanin@universite-paris-saclay.fr</footnote>

<footnote>§ alex.chin@insp.jussieu.fr</footnote>
$t_2$, the polymer stabilizes a non-trivial topological phase.

Moreover, the addition of the second bridge introduces a pair of bands that exhibit pronounced sensitivity to strain, enabling multiple continuous topological transitions characterized by band gap closures with reasonable amounts of strain. Through systematic application of strain, we elucidate the evolution of these transitions and their significant impact on the electronic and optical properties of the polymer, paving the way for novel low-bandgap organic materials with tunable topological features. In this work, we also explore the interplay between bond length alternation (BLA), bandgap, and topological phase transitions in the [n = 6] acene polymer with two bridges under compressive and tensile strain. Our findings provide critical insights into the design of ultra-low-bandgap organic polymers and open new avenues for topology-driven applications in organic optoelectronic systems.

<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematic representation of the polyacene polymer considered in this work. The polymer consists of acene monomers, each comprising six aromatic rings linked by two bridges. The polymer chains are aligned along the x-direction, forming a one-dimensional periodic structure.</figcaption></figure>

The polyacene polymer under investigation is composed of acene monomers, each consisting of six linearly aromatic rings (hexacene), connected by two ethynyl bridges (Fig. 1). The polymer chains are aligned along the x-direction, creating a one-dimensional periodic structure. The lattice constant in the equilibrium structure is $a = 6.895$ Å.

We aim to investigate the topological phase transition in a hexacene two-bridged polymer subjected to compressive and tensile strain. Strain is an effective tool for modifying the crystal structure of materials, which alters their electronic properties and, possibly, their topologi-

cal characteristics [20, 23-30]. To analyze these strain-induced topological phase transitions, we employ a series of computational methods.

## II. COMPUTATIONAL METHODS

### A. Ground-state and geometry optimization

To calculate the ground-state electronic structure properties in the framework of Density Functional Theory (DFT), the structure has been relaxed using a Gaussian basis with the CAMB3LYP exchange-correlation functional [31, 32] designed to consider the long-range corrections for carbon-based materials and the triple zeta-polarized Gaussian type basis set with real space integration tolerances of 10-10-10-25-50 and an energy tolerance of 10-12 Ha for the total energy convergence [33] in the CRYSTAL17 computational package [34, 35]. To avoid the interaction between each supercell, a large vacuum of 500 Å along nonperiodic y and z directions has been taken. The sampling of the Brillouin zone employs a uniform mesh configuration of 50 × 1 × 1 k-points, with 50 k-points along the x periodic direction and 1 k-point each along the remaining two directions. For the calculations of the GW band structure, we also achieve convergence with this same k-point configuration.

### B. Many-body interactions

To account for many-body interactions, we refined the Kohn-Sham (KS) energies using the single-shot GW ($G_0W_0$) approximation, which was implemented in YAMBO code [36, 37]. The quasiparticle equation is solved using standard approximations, i.e., the random phase approximation (RPA) and plasmon pole approximation (PPA) [38]. The ground-state wave functions and eigenvalues are computed employing the plane-wave basis set and pseudopotential approach within the Quantum ESPRESSO computational package [39, 40] using the relaxed atomic geometries extracted from the Gaussian basis set. The KS-DFT exchange-correlation functional was approximated using Perdew-Burke-Ernzerhof (PBE) [41]. The KS electronic structure was computed by setting a cutoff of 70 Rydbergs for the plane wave kinetic energy and a threshold of $10^{-10}$ Ry for the total energy, employing a norm-conserving pseudopotential. To ensure a well-defined one-dimensional system, a vacuum space of 10 Å along the non-periodic y and z directions is implemented, along with a Coulomb potential cutoff [42].

In the Supplementary Material [43], we confirm the trends in the band gap and band structure behavior under strain; additional calculations were conducted using the evGW approximation for both the equilibrium configuration and selected strained systems. These calculations
<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Band structure of the polyacene polymer under compressive strain calculated using the PBE functional. The panels show the band structure at equilibrium and under 2.9%, 4%, 6%, and 8% compressive strain, along with the corresponding projected density of states (PDOS) on the right. (b) The panels display the band structure under 2%, 3.5%, 4%, and 6% tensile strain, with the corresponding PDOS on the right side of each plot. The grey and red arrows indicate the Zak invariant ($Z_2$, where $Z_2 = 1$ represents a non-trivial topological phase and $Z_2 = 0$ a trivial phase) for the specified bands. (c) Charge density distribution on the monomer and bridge units of the polyacene polymer under compressive and tensile strain. The color bar on the right indicates the intensity of charge density across the bonds.</figcaption></figure>

further refine the band gap estimates and validate the band structure trends. The evGW (Self-consistent GW) results corroborate the findings from the PBE and $G_0W_0$ calculations, reinforcing the observed strain-induced electronic properties (Fig. S1). We have also previously shown that the use of self-consistent GW calculations (only on eigenvalues) on top of the DFT-PBE results provides extremely close results to the CAM-B3LYP functional in linear $\pi$-conjugated carbon systems [19, 44].

Additionally, we discuss the connection between the

Zak phase and the band topology of the system in the context of the Su-Schrieffer-Heeger (SSH) model, which effectively describes single-bridged polyacene polymers. We extend this formalism to a "double-stacked" four-site SSH model to represent the double-bridged polyacene polymers studied here. The supplementary material [43] (see also references [1-11] therein) also includes detailed information on the Wannier interpolation for obtaining the Zak phase, the choice of trial orbitals, and the tight-binding model used to analyze the topological
phases at different strain levels.

### III. RESULTS AND DISCUSSION

#### A. Equilibrium

At equilibrium, the [n=6] acene polymer exhibits a topologically non-trivial electronic structure, as illustrated in Panel (a) of Fig. 2. Zak phase analysis confirms that the HOMO and LUMO are located on non-trivial bands, reflecting the intrinsic topological character of the system. In Supplementary Material Fig. S6 [43] (see also references [12-13] therein), the finite energy spectrum obtained using the Wannier-interpolated tight-binding model for a system of 22 monomers at equilibrium shows the presence of midgap states, confirming the non-trivial topological character. In contrast, the HOMO-1 and LUMO+1 lie on trivial bands. This configuration serves as a critical reference point for understanding the system's response to strain and its potential for multiple topological phase transitions.

The electronic band gap is computed for equilibrium and under both compressive and tensile strains using three different methodologies: DFT-PBE, single-shot GW (G₀W₀), and self-consistent GW. The results for the band gap at equilibrium and various strain levels are summarized in Table I. DFT calculations with the PBE functional predict a band gap of approximately 0.16 eV at equilibrium, which is notably underestimated. Incorporating many-body perturbation using GW theory with DFT enhances the band gap to 0.29 eV, nearly doubling the initial value. This small gap suggests a promising potential for strain-induced band exchange and topological transition, given the proximity of the valence band maximum (VBM) and conduction band minimum (CBM).

To further understand the interplay between molecular structure and topological behavior, we analyze the charge density distributions under equilibrium and strain conditions. The charge density plots in panels (c) and (d) of Fig. 2, respectively, represent the electronic distribution of the system at equilibrium and under strain. These plots provide critical insights into electron localization and bonding characteristics, helping to infer potential structural transitions, such as the aromatic to quinonoid transformation. At equilibrium, the structure exhibits a semi-aromatic character due to charge density localization over the hexagonal rings and along the bridges. While the rings show delocalized π-electron clouds, suggesting aromaticity, the presence of charge density on the bridges connecting the polymer chains, denoted by the dashed black circle in panel (c) of Fig. 2, indicates a partial deviation from perfect aromaticity. Previous studies have shown a strong correlation between aromatic structures and trivial topological

phases, while quinonoid structures are generally linked to non-trivial phases (Z₂=1) in π-conjugated polymers [21]. In contrast, our findings present a semi-aromatic structure that exhibits a non-trivial topological phase, suggesting a more complex relationship between molecular structure and topological behavior.

To systematically investigate the effect of strain on the band gap and explore its role in driving topological phase transitions, we applied both compressive and tensile strain along the x-axis of the system. The lattice constant was progressively reduced by 2.0%, 2.9%, 4.0%, 6.0%, and 8.0% under compressive strain and increased by 2.0%, 3.5%, 4.0%, and 6.0% under tensile strain. Following the application of strain at each level, the atomic geometry was fully relaxed at the level of DFT using the CAMB3LYP functional to allow the system to reach its lowest energy configuration. It has been demonstrated the CAMB3LYP provides accurate results for charge-transfer excitations, excitations to Rydberg states, and the polarizability of long chains, as well as vibrational properties, with accuracy comparable to quantum Monte Carlo methods for systems in vacuum [19]. This structural relaxation ensures that all atomic positions are optimized and that the computed ground-state energy accurately reflects the influence of both compressive and tensile strain on the electronic properties. By following this procedure, we aim to precisely capture the strain-induced modifications to the band structure and potential transitions between different topological phases.

#### B. Compressive strain

Applying compressive strain to the system introduces significant changes in its electronic properties, profoundly influencing the band structure and wavefunction characteristics and leading to the emergence of strain-induced topological phase transitions. Zak phase analysis reveals that the [n=6] acene polymer exhibits notable shifts in its electronic structure, as illustrated in Panel (a) of Fig. 2. While the equilibrium state is characterized by the HOMO and LUMO lying on non-trivial bands, the application of compressive strain induces multiple topological transitions. Initially, the HOMO and LUMO bands move toward a trivial regime, resulting in a topological phase transition where these bands lose their non-trivial character. With increasing strain, a second transition occurs, restoring the HOMO and LUMO to non-trivial bands. These transitions highlight the intricate interplay between strain and the topological nature of the electronic structure.

As can be seen in Fig. 3, the first transition of the HOMO and LUMO characteristics occurs at 2.9% strain, where the non-trivial bands of the HOMO and LUMO and the trivial bands of the HOMO-1
<figure><img src="image_4.png" /><figcaption>FIG. 3. Charge density distribution of the HOMO and LUMO states at the edge of the Brillouin zone. Under compressive strain, a transition from non-trivial to trivial transition is observed at 2.9% strain, followed by a reversion to a non-trivial phase at 6% strain. Under tensile strain, a transition from non-trivial to trivial transition occurs at 3.5% strain, where a quasi-metallic phase is observed. Positive values denote the tensile strain.</figcaption></figure>

and LUMO+1 cross as their band gaps increase and decrease, respectively. At this critical strain, the trivial bands become the valence band maximum (VBM) and conduction band minimum (CBM), causing the HOMO and LUMO to become trivial. However, this is not a conventional topological phase transition but rather a strain-induced band rearrangement. As compressive strain increases further, notably at 6%, a topological phase transition occurs. This transition is characterized by a band inversion at the Fermi level (see Fig. 3), where the HOMO and LUMO exchange their characteristics, transitioning from trivial back to non-trivial bands. This inversion marks a critical point in the system's electronic behavior, signifying the re-establishment of non-trivial topology. These two distinct transitions, one from non-trivial to trivial and the other from trivial to non-trivial, highlight the complex interplay between strain and the polymer's electronic structure. Therefore, with compressive strain, we observe two distinct phase transitions, the first unusual one from non-trivial to trivial due to the crossing of the neighboring bands, and the second usual one from trivial to non-trivial due to the band inversion at the Fermi level.

This trend matches the band gap behavior as a function of BLA displayed in panel (a) of Fig. 4 (which follows the same correlation between band gap and strain in π-conjugated systems) and exhibits a non-monotonic response: an initial increase from 0.16 eV to a maximum of 0.5 eV at 2.9% strain, followed by a sharp drop to 0.076 eV at 6% strain. This drop in the band gap, followed by

a partial recovery at higher strains, creates a V-shaped profile, indicative of a topological phase transition (see panel (a) of Fig. 4), analogous to observations in other 1D conjugated systems [20]. The V-shaped response of the band gap as a function of compressive strain suggests that the applied strain drives the system through a critical point, where the nature of the electronic states changes, leading to the observed inversion at 6%. Notably, at -2.9% strain, where the band gap reaches its maximum due to the crossover of the aforementioned bands, a non-conventional topological phase transition occurs as a result of strain-induced modifications in the band structure. In this unusual transition, the band gap does not undergo the typical closing and reopening process, and thus, a V-shaped behavior is absent.

Consequently, the correlation between the band gap behavior as a function of BLA or strain and the observed wavefunction inversion highlights the complex influence of strain on the topological properties of the polymer system.

Under compressive strain, the polymer exhibits a notable evolution in its electronic bandgap at both the DFT and GW levels. The maximum value is observed at 2.9%, reaching 0.5 eV with DFT and 1.31 eV with DFT + many-body perturbation, respectively. As a remark, at equilibrium, the PBE and GW band gaps are 0.16 eV and 0.29 eV, respectively. Under 6% compressive strain, where the PBE bandgap reduces to 0.076 eV, the GW bandgap remains larger at 0.59 eV, a greater difference than what is observed at equilibrium. This
<figure><img src="image_5.png" /><figcaption>FIG. 4. (a): The black line shows the bond length alternation (BLA) as a function of strain, while the red line depicts the corresponding band gap. (b): Variation in bond lengths of the polyacene polymer under compressive strain. The bond lengths $l_1$, $l_2$, $l_3$, and $l_4$ are illustrated in Fig. 1. (c): Energy levels of the HOMO and LUMO states as a function of strain. The red arrow at -2.9% and the dotted oval at +3.5% indicate the non-trivial to trivial topological phase transition, while the dotted oval at -6% represents the trivial to non-trivial transition.</figcaption></figure>

discrepancy can be attributed to the emergence of flat bands in the HOMO-1 and LUMO+1 states. These flat bands indicate strong localization of electronic states, which reduces the interaction between adjacent energy levels (HOMO vs. HOMO-1 and LUMO vs. LUMO+1). Additionally, the increasing separation of these bands from an energetic perspective weakens the screening effect, leading to enhanced Coulomb interaction in the localized states. As a result, the quasiparticle (QP) bandgap remains relatively large.

Panel (a) of Fig. 2 presents the band structures of equilibrium and strained systems. At a strain of 2%, the HOMO retains its bandwidth compared to the equilibrium, while LUMO's is slightly affected. At 2.9% strain, the HOMO-1 becomes the VBM. The conduction bands exhibit degeneracy at the edge of the Brillouin zone, which is removed at 4.0% strain, where LUMO+1 becomes the CBM. At higher strain values, the HOMO-1 and LUMO+1 exhibit a flat band, while the HOMO and LUMO display wider bands. We computed the Partial Density of States (PDOS) to analyze the orbitals' characteristics near the Fermi surface. As shown in panel (a) of Fig. 2, it is revealed that the carbon atoms in both the rings connected to the bridges and central rings make the primary contribution near the Fermi level and are significantly impacted by strain.

Strain not only affects the electronic properties of the polymer but also induces significant structural changes, as evidenced by the variations in bond lengths. We explored the bond lengths labeled as $l_1$, $l_2$, $l_3$, and $l_4$ in Fig. 1 and panel (b) of Fig. 4, which, in equilibrium, are 1.2071 Å, 1.4156 Å, 1.3942 Å, and 1.4346 Å, respectively. Upon applying compressive strain, we observe variations in the bond lengths. As depicted in panel (b) of Fig. 4, the bond length $l_1$ exhibits only minimal variation,

ranging from 1.2071 Å to 1.1949 Å, reflecting its relative rigidity and reduced responsiveness to strain that can be attributed to the large force of the $-C≡C-$ triple bond. In contrast, the bond length $l_2$ responds most significantly to strain, contracting from 1.4156 Å to 1.2731 Å as the compressive strain increases from 0% to 9%. This substantial decrease in $l_2$ demonstrates its sensitivity to mechanical forces, likely due to its critical structural role in the polymer. The bond length $l_3$ fluctuates moderately between 1.3942 Å and 1.4407 Å, suggesting an intermediate level of sensitivity to strain.

Moreover, we examined the effect of compressive strain on the polymer by analyzing the bond length alternation (BLA) and the electronic band gap ($E_g$). Panel (a) of Fig. 4 depicts the behavior of BLA as a function of strain. The BLA values, shown in Fig. 4 (a) (black line), decrease with increasing compressive strain, from 0.21 Å at equilibrium to 0.07 Å at 9% strain. The band gap $E_g$, represented by the red line in panel (a) of Fig. 4, exhibits a non-monotonic behavior. The bandgap of the system at equilibrium (0% strain) is 0.16 eV. Under compressive strain, the bandgap initially increases, reaching a maximum of 0.5 eV at 2.9% compressive strain before decreasing to 0.26 eV at 9% strain. The minimum value of 0.076 eV is observed at 6% compressive strain. To gain a deeper understanding, we have computed the decomposed charge densities for the HOMO and LUMO, as shown in Fig. 3. At equilibrium, the HOMO at the edge of the Brillouin zone exhibits an anti-bonding nature between the two carbon atoms forming the triple bond ($C≡C$). Under compressive strain, This leads to stabilization of the HOMO energy level, while the LUMO becomes destabilized (see panel c of Fig. 4). Consequently, the band gap increases up to 2.9%, where a non-trivial to trivial transition occurs. Beyond this point, the HOMO
adopts a π-bonding character, resulting in a reduction in the band gap. The eventual narrowing of the band gap beyond 6% compressive strain suggests the polymer approaches a near-metallic state. This behavior is in accordance with the findings reported by J. L. Brédas [45, 46], where the band gap was observed to decrease linearly with increasing quinoid-cumulene character.

Under compressive strain, the electronic structure of the polymer undergoes remarkable changes, as evidenced by the charge density plots in panel (c) of Fig. 2. These changes shed light on the evolving electron localization and bonding patterns, reflecting the system's adaptation to external mechanical strain. As shown in panel (c) of Fig. 2, compressive strain drives a redistribution of charge, particularly between the bridge and monomer units, signaling a shift from a semi-aromatic structure at equilibrium to a quinonoid electronic configuration. The bridging units transition from ethynyl (–C≡C–) to cumulenic (C=C=C) bonding, accompanied by a pronounced localization of charge within the monomer rings. A more detailed analysis of the charge density evolution at each strain level, illustrating the progressive transition from a semi-aromatic to a quinonoid or aromatic structure, is provided in Fig. S8 in the Supplementary Material [43].

As shown in panel (a) of Fig. S8, at 2.9% strain, the charge density becomes increasingly localized between the bridges and the polymer chains, with more pronounced asymmetry across the structure. While the central rings retain much of their aromatic character, the localized charge suggests an initial response to strain, hinting at an impending transition toward a more quinonoid configuration. Above 4% strain, significant changes are observed in the charge density distribution. The bridges between the polymer chains exhibit greater charge localization, particularly around the connecting bonds, indicating a deviation from the typical delocalized electron pattern seen in ethynyl (–C≡C–) structures. Within the central rings, the electron cloud is less uniformly distributed, which suggests that the semi-aromatic delocalization is breaking down, leading to an aromatic-to-quinonoid transition. The localization of charge in this region becomes a key marker of structural change in response to strain. As strain increases to 6%, the transition becomes much more pronounced. The charge density within the central rings of the polymer becomes localized, a signature feature of quinonoid structures where electron delocalization is reduced. The previously delocalized electron cloud in the aromatic state is now replaced by distinct regions of charge concentration around specific bonds, marking the system's progression toward a quinonoid configuration. Additionally, the bridges show a shift from the ethynyl (–C≡C–) to a cumulenic (C=C=C) structure, further confirming that significant changes in both the electronic and geometric structure is occurring at this strain level. The electron density is heavily concentrated around certain bonds in the central rings, indicating that the system is no longer

semi-aromatic. At 8% strain, the localization of charge density becomes even more pronounced, confirming the formation of a quinonoid structure. The bridges show a fully cumulenic structure, with the electron density distributed consistently with cumulenic bonding. The transition from a semi-aromatic, ethynyl-bridged polymer to a quinonoid, cumulenic structure is complete at this strain level.

The BLA in a π-conjugated system displays behavior similar to strain direction in the sense that compressive strain, where the bond lengths of the system decrease systematically, as shown in panel (b) Fig. 4, leads to a progressive decrease in BLA. This is in agreement with the application of compressive strain in poly(PENT-yne) and poly(ANTH-yne) [20]. Therefore, the band gap simultaneously exhibits a V-shaped behavior as a function of strain and BLA, which is a signature of the conventional topological phase transition. The fluctuations in BLA depend on the relative changes between $l_1$ and $l_2$. Since, at equilibrium, $l_1$ has the ethynyl (–C≡C–) form, it resists change; consequently, variations in $l_2$ have a more significant effect on BLA. Under compressive strain, the fluctuations in BLA can be attributed to the transition from semi-aromatic to quinoid.

Our results highlight the fact that the correlation between band gap and BLA in π-conjugated systems under external mechanical strain is exactly analogous to the correlation between strain and band gap [20]. However, in other polymers such as PPD and TTD, the correlation between band gap and BLA follows the opposite trend, decreasing compared to the strain–band gap correlation. [21].

<table><caption>TABLE I. Electronic band gap values (in eV) under various levels of tensile and compressive strain, as calculated using different methods: PBE and G₀W₀. The band gap values computed using the self-consistent GW method are provided in Table SI.</caption><thead><tr><th>System</th><th>PBE (eV)</th><th>G<sub>0</sub>W<sub>0</sub> (eV)</th></tr></thead><tbody><tr><td>+6%</td><td>0.127</td><td>0.43</td></tr><tr><td>+4%</td><td>0.034</td><td>0.33</td></tr><tr><td>+3.5%</td><td>0.004</td><td>0.25</td></tr><tr><td>+2%</td><td>0.056</td><td>0.05</td></tr><tr><td>Equilibrium</td><td>0.16</td><td>0.29</td></tr><tr><td>-2%</td><td>0.26</td><td>0.59</td></tr><tr><td>-2.9%</td><td>0.5</td><td>1.31</td></tr><tr><td>-4%</td><td>0.49</td><td>1.16</td></tr><tr><td>-6%</td><td>0.076</td><td>0.52</td></tr><tr><td>-7%</td><td>0.12</td><td>0.67</td></tr><tr><td>-8%</td><td>0.19</td><td>0.82</td></tr><tr><td>-9%</td><td>0.26</td><td>1.00</td></tr></tbody></table>

### C. Tensile strain

The application of tensile strain shifts the system toward a trivial regime, causing the HOMO and LUMO
bands to transition from non-trivial to trivial character. This strain leads to a rapid decrease in the band gap, with the PBE functional predicting a sharp reduction that brings the system to a quasi-metallic state (0.004 eV) at 3.5% strain as observed in poly(PENT-yne) [20]. The observed decrease in the band gap is linked to the anti-bonding nature of the HOMO in the bridges. Once the non-trivial to trivial transition occurs, the band gap gradually increases (see panel a of Fig. 4). Moreover, it is evident that the band gap as a function of BLA or strain follows a V-shaped trend due to the closure and reopening of the band gap at 3.5% tensile strain, corresponding to a conventional non-trivial to trivial topological phase transition. To further analyze this transition at higher tensile strain, the finite energy spectrum has been computed. In Fig. S7 in the Supplementary Materials [43], the midgap state observed at equilibrium (indicative of a non-trivial phase) is clearly absent, confirming the transition to a trivial phase.

The bandgap closure forms a quasi-metallic phase, as evidenced by continuous decomposed charge density across the bridges. This finding highlights the emergence of a fully cumulenic structure, providing critical insights into the polymer's electronic behavior, as displayed in Fig. 3. Beyond this point, the bandgap begins to recover slightly, increasing to 0.127 eV at 6% strain.

The effect of tensile strain on the band structure is illustrated in panel (b) of Fig. 2. Unlike the compressive strain scenario, the tensile strain does not lead to band crossings. The HOMO and LUMO distributions remain largely unaffected, as indicated by the PDOS plots, while the separation between HOMO/LUMO and HOMO-1/LUMO+1 increases. This change corresponds to the structural expansion under tensile strain, as illustrated in panel (b) of Fig. 4, which influences the system's electronic characteristics. Under tensile strain, $l_1$, which represents the bond between two key atoms in the conjugated backbone, increases from 1.2071 Å at equilibrium to 1.2424 Å at 8% tensile strain. This gradual lengthening of $l_1$ signifies a weakening of the bond strength. The increase in bond length for $l_1$ under tensile strain can be understood as a response to the reduced $\pi$-overlap between the atoms due to the applied tensile strain, which stretches the conjugation. Similarly, $l_2$, which is typically associated with the single bond adjacent to the bridge, shows a more pronounced elongation. Starting from 1.4156 Å at equilibrium, $l_2$ increases to 1.5506 Å at 8% strain. The bond lengths $l_3$ and $l_4$ also follow the expected trend. The $l_3$ bond, which represents another key $\pi$-conjugated bond, increases from 1.3942 Å at equilibrium to 1.4089 Å at 8% strain. This more modest increase compared to $l_2$ is consistent with the fact that $l_3$ typically remains closer to an ideal double bond, retaining more aromatic character even under strain. On the other hand, $l_4$, similar to $l_2$, shows a more significant elongation, increasing from 1.4364 Å to 1.5032 Å. These findings highlight the varied effects of strain on different

bond lengths within the polymer, offering valuable insights into how strain influences both the structural and electronic properties of the system.

The tensile strain produces a clear and consistent elongation of bonds, which directly increases the BLA. This elongation, particularly in the $l_1$ and $l_2$ bonds, reflects the retention of the ethynylene-like structure under tensile conditions (see Fig. 4 (a) and (b)). As shown in panel (c) of Fig. 2 and panel (b) of Fig. S8, under tensile strain, the charge density remains delocalized across the acene rings, indicating the preservation of the semi-aromatic character of the polymer. Even at 6% tensile strain, the charge density maintains its uniformity across the conjugated backbone, with the ethynylene bridges between the acene rings largely unaffected. This observation is consistent with the gradual increase in BLA and the reduction in the HOMO-LUMO gap under tensile strain.

## IV. CONCLUSION

In conclusion, this study presents a comprehensive investigation of strain-induced topological phase transitions in [n=6] conjugated polyacene polymers, focusing on the interplay between mechanical deformation and electronic properties. Through first-principles calculations, we demonstrate that both compressive and tensile strains significantly impact the band gap, charge density distribution, and bond length alternation (BLA). At equilibrium, the system is shown to be in a non-trivial topological phase ($Z_2$ = 1), with a semi-aromatic structure. Under compressive strain, the polymer experiences a topological phase transition from non-trivial to trivial at 2.9%, followed by a reversion to a non-trivial phase at 6%, exhibiting a quinoid structure. This demonstrates a sequence of multiple topological phase transitions. Conversely, tensile strain induces a transition from a non-trivial to a trivial phase at 3.5%, leading to a quasi-metallic state characterized by bandgap closure and an enhanced aromatic character. Beyond this critical strain, further stretching reopens the bandgap, indicating a second phase transition. These results underscore the pivotal role of strain in tuning both the electronic structure and topological characteristics of conjugated polymers. The non-monotonic behavior of the band gap highlights the potential of these materials for flexible, adaptive organic electronic devices. This work advances our understanding of topological materials and paves the way for designing low-bandgap polymers with tailored properties for applications in optoelectronics and next-generation semiconductors.

## 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-CE50-0001-01) 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). Special thanks to the MCMeSU, which is built with high-performance hardware to address the numerical simulation and intensive computing needs of the research community.

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# Multiple Strain-Induced Topological Phase Transitions in 1D Polyacene Polymers: A First-Principles Study

M. Mosaferi¹,* M. Aktas²,† D. Romanin²,‡ and A. W. Chin¹§

¹Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252 Paris, France and

²Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

(Dated: May 13, 2025)

## I. COMPUTATIONAL METHODS

### A. Self-consistent GW

To corroborate the trends in band gap evolution under varying strains as predicted by PBE and G₀W₀ calculations, we performed additional self-consistent GW (evGW) calculations. These results further validate the observed electronic structure modifications under equilibrium and strained conditions. The evGW calculations were carried out up to three iterations (G₁W₁, G₂W₂, and G₃W₃) to ensure convergence of the band gap and to track the band structure's response to strain, with particular focus on the evolution and distribution of the valence band maximum (VBM) and conduction band minimum (CBM). These findings affirm the consistency of PBE and G₀W₀ predictions in band structure while showcasing the importance of evGW in refining band gap estimates.

<figure><img src="image_6.png" /><figcaption>FIG. 1: The upper panel depicts the band gap evolution across three steps of self-consistent GW calculations (G₁W₁, G₂W₂, and G₃W₃) for the equilibrium structure, as well as under -2.9% and -4.0% compressive strain and +3.5% tensile strain. The lower panel shows the evG₃W₃ band structure for the equilibrium and strained systems, demonstrating strain-induced changes in the band structure consistent with PBE and G₀W₀ results.</figcaption></figure>

Table I presents the bandgap values obtained at equilibrium and under varying strain conditions using the self-consistent GW method. Starting with G₁W₁, the bandgap values systematically refine through G₂W₂ and converge by G₃W₃. For example, at equilibrium, the bandgap refines from 0.37 eV (G₁W₁) to 0.39 eV (G₂W₂) and reaches 0.40 eV (G₃W₃), showcasing the incremental yet consistent improvement with each step. Similar trends are observed under strain.

<table><caption>TABLE I: Bandgap refinement at equilibrium and under varying strain conditions using the self-consistent GW method.</caption><thead><tr><th>System</th><th>G<sub>1</sub>W<sub>1</sub> (eV)</th><th>G<sub>2</sub>W<sub>2</sub></th><th>G<sub>3</sub>W<sub>3</sub></th></tr></thead><tbody><tr><td>+3.5%</td><td>0.52</td><td>0.57</td><td>0.59</td></tr><tr><td>Equilibrium</td><td>0.37</td><td>0.39</td><td>0.40</td></tr><tr><td>-2.9%</td><td>1.65</td><td>1.77</td><td>1.79</td></tr><tr><td>-4%</td><td>1.45</td><td>1.55</td><td>1.60</td></tr></tbody></table>

<footnote>* mosaferi@insp.jussieu.fr</footnote>

<footnote>† melih.aktas@etu-upsaclay.fr</footnote>

<footnote>‡ davide.romanin@universite-paris-saclay.fr</footnote>

<footnote>§ alex.chin@insp.jussieu.fr</footnote>
## B. Band Gap with B3LYP Functional

To address concerns regarding the underestimation of the bandgap using the PBE functional, we performed additional calculations employing the hybrid B3LYP functional, as shown in Fig. 2. As expected, the B3LYP functional yields a systematically larger bandgap compared to PBE; however, the overall trend of bandgap closure and reopening under strain remains consistent. This result confirms that the observed topological phase transitions are intrinsic to the system’s electronic structure rather than an artifact of the exchange-correlation functional. In all cases, the topological nature of the transitions was verified by computing the Zak invariant ($Z_2$), confirming the existence of multiple topological phase transitions. These additional calculations demonstrate that while absolute bandgap values are functional-dependent, the fundamental topological characteristics of the system remain unchanged.

<figure><img src="image_7.png" /><figcaption>FIG. 2: Band gap evolution under strain for different functionals (PBE, B3LYP, and GW). The band gap trend, featuring closure and reopening, is consistent across all methods, confirming the topological phase transition. B3LYP and GW yield larger band gaps compared to PBE, with slight variations in critical strain values. The Zak invariant ($Z_2$) was computed in all cases to verify the topological phase transitions.</figcaption></figure>

## C. Topological invariants

To obtain the Zak invariant of the polymers, Wannier interpolation [1, 2] has been performed, with the toolkit Wannier90 [3], on the wave functions and the eigenvalues obtained with Quantum ESPRESSO.

More precisely, the convenient way to carry out the Wannier interpolation is to provide the system with a set of initial guesses of the orbitals responsible for the

selected subset of bands. For this purpose, we project the energy bands onto their orbital characters:

$$|\psi_{nk}\rangle = \sum_j C_j^{nk} |\phi_j\rangle \quad (1)$$

where $|\psi_{nk}\rangle$ is the Bloch function of the $n^{th}$ band corresponding to the canonical Bloch Hamiltonian $e^{-ikx}He^{ikx}$ and $|\phi_j\rangle$ are the orbitals of the system. We obtain the probability amplitudes $|C_j^{nk}|^2$.

From Fig. 3 it is possible to observe that the main orbitals responsible for the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of the double-bridged polyhexacene polymer are the $p_z$ orbitals on the Carbon atoms connected by the ethynylene bridges. Therefore, we use the four $p_z$ orbitals centered on these atoms as starting guesses for the Wannier interpolation (Fig. 4).

Non-self-consistent calculations were carried on a grid of 20 × 1 × 1 k-points and the $p_z$ orbitals of the four Carbon atoms connected by the ethynylene bridges were used as the trial orbitals as they are the main orbitals giving rise to the bands closest to the Fermi level. For each strain level, we construct a four-site tight-binding model by using the maximally localized Wannier functions as a basis and calculate the Zak phase over each band to determine their topological phase as described below in more detail.

## II. ZAK PHASE AND THE 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 [4]. This formalism can be generalized to a four site "double stacked" SSH model which represents double-bridged polyacene polymers.

### A. Su-Schrieffer-Heeger Model

The SSH model is a one-dimensional tight-binding model originally introduced to describe polyacetylene [5]. Due to the dimerized nature of the chain, it consists of orbitals with the same on-site energies and alternating hopping parameters (Fig. 5).

The real space Hamiltonian is given by:

$$\hat{H} = t_1 \sum_n c_{nB}^\dagger c_{nA} + t_2 \sum_n c_{(n+1)A}^\dagger c_{nB} + \text{H.c.} \quad (2)$$
<figure><img src="image_8.png" /><figcaption>FIG. 3: Band structure of the double-bridged polyhexacene polymer with the orbital projections. 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 shapes are proportional to the sum of the probability amplitudes of each set of orbitals.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 4: Wannierisation of the strain-free system: a) Wannier interpolated band structure with the horizontal gray lines corresponding to the inner and outer energy windows. b) 4-site SSH model constructed with MLWFs.</figcaption></figure>

where $c_{nA}^{\dagger}$ ($c_{nA}$) and $c_{nB}^{\dagger}$ ($c_{nB}$) are creation (annihilation) operators of fermions on sublattices A and B
<figure><img src="image_10.png" /><figcaption>FIG. 5: Schematic representation of SSH in each topological phase, with the thick lines corresponding to stronger bonds. Left: $t_1 > t_2$ and consequently the Wannier center is located at the center of the intra-cell bond producing $\mathbb{Z}_2 = 0$, a trivial SSH. Right: $t_1 < t_2$ and consequently the Wannier center is located at the center of the inter-cell bond producing $\mathbb{Z}_2 = 1$, a non-trivial SSH.</figcaption></figure>

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. The equal on-site energies (cast in the form of an identity matrix) are neglected as they do not have any effect on the physics.

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 a finite chain in the non-trivial phase [6, 7].

## B. $\mathbb{Z}_2$ 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) and it is intimately connected with the band topology:

$$ \phi_n = i \int_{\text{BZ}} \langle u_{nk} | \nabla_k | u_{nk} \rangle d\mathbf{k} \quad (3) $$

where $\nabla_k$, $|u_{nk}\rangle$ and $n$ are the gradient operator, the cell-periodic Bloch function and the band index, respectively. The integral is carried out in a non-contractible loop over the BZ torus.

The average position of a Wannier function positioned in the unit cell at the origin of the crystal ($R = 0$), $\langle \mathbf{r} \rangle = \langle w_0 | \mathbf{r} | w_0 \rangle$, called the Wannier center can be cast in the following form [8] (we omit the band index without loss of generality):

$$ \begin{aligned} |w_0\rangle &= \frac{V}{(2\pi)^D} \int_{\text{BZ}} d\mathbf{k} |\psi_{\mathbf{k}}\rangle \\ &= \frac{V}{(2\pi)^D} \int_{\text{BZ}} d\mathbf{k} e^{i\mathbf{k}\cdot\mathbf{r}} |u_{\mathbf{k}}\rangle \end{aligned} \quad (4) $$

$$ \begin{aligned} |\mathbf{r}|w_0\rangle &= \frac{V}{(2\pi)^3} \int_{\text{BZ}} d\mathbf{k} (-i\nabla_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}) |u_{\mathbf{k}}\rangle \\ &= i \frac{V}{(2\pi)^D} \int_{\text{BZ}} d\mathbf{k} e^{i\mathbf{k}\cdot\mathbf{r}} (\nabla_{\mathbf{k}}|u_{\mathbf{k}}\rangle) \end{aligned} \quad (5) $$

$$ \langle w_0 | \mathbf{r} | w_0 \rangle = i \frac{V}{(2\pi)^D} \int_{\text{BZ}} d\mathbf{k} \langle u_{\mathbf{k}} | \nabla_{\mathbf{k}} | u_{\mathbf{k}} \rangle \quad (6) $$

where $V$ and $D$ are the volume of the unit cell and the dimensions of the system, respectively. We used the conjugate relation between the position and the momentum operator in equation (5).

From equations (3) and (6), one gets the relation:

$$ \langle \mathbf{r} \rangle_n = \frac{V \phi_n}{(2\pi)^D} \quad (7) $$

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

$$ \langle x \rangle_n = \frac{\phi_n}{2\pi} \quad (8) $$

This relation between the Zak phase and the Wannier center is pivotal to classifying the topology. The Wannier centers in one-dimensional crystals are usually located at the inversion centers. In SSH, they are located at the centers of the bonds in between the sites. Moreover, one can show that for both the valence and the conduction band, the Wannier centers are always located at the center of the intra(inter)-cell bond in the trivial (non-trivial) phase [7, 9].

Therefore, by calculating the Zak phase of either couple of bands (as they have the same value) we
determine where the Wannier center is located and consequently the topological phase (see Fig. 5).

The Zak phase is continuously dependent on the Wannier center, therefore the choice of the unit cell. When one Wannierises the polyacene polymers, one obtains SSH-like models with the A and the B sites not necessarily at the exact same positions from one polymer to the other. 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, i.e the Wannier center of a topologically trivial SSH [10]:

$$Z_2 := \frac{\phi_n - 2\pi \langle x \rangle_n^{\text{triv}}}{\pi} \qquad (9)$$

with this definition, we get $Z_2 = 0$ for a trivial SSH since the Zak phase produces the intra-cell bond center as well and with a clever choice of the denominator we get $Z_2 = 1$ for a non-trivial SSH as the Wannier center shifts by half a unit cell from the trivial case.

### III. NUMERICAL 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 ([6], chapter 3.4):

$$\phi_n = - \operatorname{Im} \ln \left[ \langle u_n(k_0) | u_n(k_1) \rangle \langle u_n(k_1) | u_n(k_2) \rangle \right. \\ \left. \dots \langle u_n(k_{N-1}) | e^{-iG \cdot r} | u_n(k_0) \rangle \right] \qquad (10)$$

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 employed to ensure smoothness at the borders of BZ. The

off-diagonal terms of the position operator r was ignored as they were three orders of magnitude smaller than the diagonal terms. The calculations were done with the tight-binding Python library PythTB [11].

### IV. MIDGAP STATE IN FINITE POLYMER

Midgap states appear in finite systems due to the presence of boundaries, whereas in infinite systems, we focus on the bulk properties of topological materials. Edge states arise as a direct consequence of the bulk-boundary correspondence (BBC), which posits that the topological invariants of the bulk (e.g., winding number or Chern number) dictate the existence and robustness of edge states in finite systems with boundaries [12]. However, in an infinite system, there are no boundaries and, consequently, no edge states. Moreover, the presence or absence of edge states serves as a key criterion for distinguishing the nontrivial from the trivial topological phase [13].

Regarding the correlation between the density of states and topological phases, we computed the finite energy spectra using the Wannier-interpolated tight-binding model for a system consisting of 22 monomers (unit cells) in both equilibrium and under 6% of tensile strain as an example for the non-trivial and the trivial phases, as shown in Fig. 6 and Fig. 7. The equilibrium case clearly exhibits midgap states, indicating a non-trivial topological phase, while the 6% tensile strain case shows their absence, suggesting a transition to a trivial phase. The comparison between equilibrium and 6% tensile strain cases strongly supports that the midgap states are topologically driven, and their disappearance under tensile strain further confirms the phase transition.

### V. CHARGE DENSITY

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<figure><img src="image_11.png" /><figcaption>FIG. 6: Finite energy spectrum obtained using the Wannier-interpolated tight-binding model for a system of 22 monomers at equilibrium with the wave functions of the midgap states at the right panel. The presence of midgap states indicates a non-trivial topological phase.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>FIG. 7: Finite energy spectrum obtained using the Wannier-interpolated tight-binding model for a system of 22 monomers for 6% tensile strain with the wave functions of the midgap states at the right panel. The absence of midgap states indicates a trivial topological phase.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 8: (a) Charge density distribution on the monomer and bridge units of the polyacene polymer under compressive strain. The aromatic ethynylene and quinoid cumulenic structures are highlighted in the color bar at the bottom. (b) Charge density distribution on the monomer and bridge units of the polyacene polymer under tensile strain. The color bar on the right indicates the intensity of charge density across the bonds.</figcaption></figure>