Topological phases of polyacenes

The gist The introduction of Su-Schrieffer-Heeger model has led to a major breakthrough in the area of one-dimensional topological insulators, even though this model was primarily formulated on an organic polymer called trans-polyacetylene in order to explain its anomalous conductivity This study introduces five tight-binding models formulated on polyacene where exotic topological behavior has been observed

Model Structures and Isomers

The investigation focuses on the topological properties of polyacenes, which are polycyclic aromatic hydrocarbons regarded as the 1D analogue of two-dimensional graphene Two geometric isomers primarily available are cis-polyacene (c-pol) and trans-polyacene (t-pol) Although the tight-binding band structures of cis-polyacene and trans-polyacetylene are found to be the same, their topological characters are totally opposite The trans-polyacetylene is nontrivial as it exhibits a topological phase with nonzero winding number, while the cis-polyacene is topologically trivial

Topological Character of Trans-Polyacene Models

The analysis of the t-pol structure shows that it is nontrivial as it exhibits topological phase with nonzero winding number The winding number for this four-band system can be expressed as ν = 1/π Z 2π0 dk v + weik = 2w 2/v squared − w 2− 2w squared v 2−w squared, if v > w, 2v 3/w2−v2, if v w, 1, if v The t-pol hosts a nontrivial topological phase with ν = 2 when −1 Experimental demonstration of these edge states has been accomplished in photonic6 and acoustic waveguides,8 mechanical oscillator9, water-wave channel10, system with Rydberg atoms emulating SSH model7, Bose-Einstein condensate of 87Rb atom into a 1D optical superlattice potential11, through quench dynamics of the system of ultracold atoms12, and topolectrical circuits13 In the region −1 1<ref:2603.25324#pg5>

Modification of Trans-Polyacene for Nontriviality

To induce topological nontriviality in the cpol structure, an additional intracell hopping term has been considered, referred to as cb-pol<ref:2603.25324#pg6> This model exhibits topological phases characterized by nonzero values of Z as well as the existence of nonzero-energy edge states<ref:2603.25324#pg6> In the tb-pol structure, introducing the additional intracell hopping term like the cb-pol structure yields a pair of flat bands and it always remain topologically nontrivial through out the parameter regime<ref:2603.25324#pg7> The resulting system yields a pair of flat bands and it always remain topologically nontrivial through out the parameter regime, in contrast to the t-pol which is found topological in a definite region<ref:2603.25324#pg7>

Properties of Cis-Polyacene Models

The cis-polyacene model (c-pol) preserves TRS, CS and PHS but does not preserve the IS unlike the t-pol The winding number in this can be expressed as ν = i vw/π Z 2π0 dk sin(k) v squared + w squared + 2vw cos (k) = 0 Although the system is non-topological, it exhibits coexistence of zero- and nonzero-energy edge states in the region −1 < v/w < +1 The presence of nonzero-energy edge states in this case actually corresponds to this real space MS of the Hamiltonian

Modified Cis-Polyacene Models

The cb-pol structure restores the inversion symmetry in the k-space and acquires the intracell rotational symmetry in the real space In order to induce nontriviality within c-pol, an additional intracell hopping path connecting A and D sublattice points has been introduced, referred to as cb-pol The Hamiltonian for this structure is given by Hcb = Hc + v X N j=1 a†j dj + h.c., (15) which satisfies the relation M(a↔d) (b↔c)−1 Hcb M(a↔d) (b↔c) = Hcb The system exhibits true band gap when v = 1/4, w = 1, as depicted in (a) with a green shade

Modified Cis-Polyacene Models with Intercell Hopping

The cn-pol model is introduced by adding four new intercell hopping terms to the c-pol structure The resulting model possesses all the symmetries except the MS cnpol exhibits three different nontrivial phases with winding numbers, ν = −2, 2, 4 The system hosts three nontrivial phases with ν = −2, 2, 4 when u = 0.7 and v = 0.

1. textbfIncreased Topological Phase Classification Capability: The improved AI can distinguish between three different topological phases in polyacene models characterized by winding numbers ν = -2, 2, and 4 based on the hopping parameters (w and u). This allows it to accurately predict whether a given model configuration will exhibit the topological phase with ν = 4 or ν = -2 as shown in Fig. 21 (b) and Fig. 23 (b).

2. textbfAccurate Topological Invariant Calculation: The system can calculate the Zak number (Zak phase per π), Zn, for all four bands in multiband systems, distinguishing between the standard nontrivial topological phase where Zak number for all bands has been specified uniquely and the anomalous topological phase where topological invariant of two bands are undefined.

3. textbfEnhanced Edge State Characterization: The improved system can quantify the localization of edge states by calculating the Inverse Participation Ratio (Ipr) and correlating its high values with specific topological regimes, such as when zero-energy edge states are highly localized in the nontrivial phase.

4. textbfModel Comparison and Symmetry Analysis: The AI can compare different polyacene structures (t-pol, c-pol, cb-pol, cn-pol) by analyzing the preservation of symmetries listed in Table I, specifically identifying cb-pol is the only model which possesses every symmetry versus others that break one symmetry.

5. textbfPredictive Phase Transition Mapping: The system can map out phase transition points in parameter space by identifying where band gap vanishes (e.g., at v/w = ±1/2 for tb-pol) or where topological invariants change, as illustrated by the vertical dashed lines in Fig 8 and Fig 21.

The introduction of Su-Schrieffer-Heeger model has led to a major breakthrough in the area of one-dimensional topological insulators, even though this model was primarily formulated on an organic polymer called trans-polyacetylene in order to explain its anomalous conductivity. In this study, a group of seven tight-binding models has been introduced which are formulated on another organic polymer called polyacene, where a variety of topological phases have been observed in six models. Topological properties of the most common geometric isomers known as cis-polyacene, and trans-polyacene have been investigated along with five additional modified polyacene structures. Although their geometric structures differ by several symmetries, tight-binding band structures of cis-polyacene and trans-polyacene are found the same, and again their topological characters are found totally opposite. The trans-polyacene is nontrivial as it exhibits topological phase with nonzero winding number, while the cis-polyacene is topologically trivial, although both the structures adhere to the same set of symmetries required for the topological character of BDI class. However, cis-polyacene possesses additional mirror symmetry in the real space but lacks inversion symmetry. Additional two modified structures of trans-polyacene and three modified structures of cis-polyacene have been considered in order to induce the nontrivial topology. Pair of flat bands is found both in modified trans-polyacene and cis-polyacene structures such that they are topologically nontrivial for trans-polyacene while trivial for cis-polyacene. Topological character of all the models is different.