Wilson polygons and the topology of zero-dimensional systems

TL;DR

This work shows that zero-dimensional systems can host non-trivial topology analogous to higher-dimensional materials by exploiting discrete rotational symmetry and defining topological indices with discrete Wilson polygons. The authors introduce a $Z_2$ index, $\mathcal{Z}_{2,N}$, computed from a discrete Berry phase along Wilson polygons, and demonstrate topological transitions in two 0-D molecules, [m]-CPP and [m]-ITN, as the inter-unit coupling $t'/t$ is tuned. They validate the framework with tight-binding models and full-band DFT (ORCA, B3LYP/STO-3G), mapping discrete molecular orbitals onto continuous bands (PPP and PITN) and correlating MO crossings with $\mathcal{Z}_{2,N}$ changes; some crossings are trivial depending on the molecular structure and symmetry. They further show that in composite nanohoops formed by topologically distinct segments, a localized boundary state emerges at the interface, with its existence and penetration depth set by the relative $Z_2$ indices and tunable via $t'/t$. The results provide a practical route to design and detect topological phases in finite molecular systems and extend topological band theory to zero-dimensional materials.

Abstract

We show that zero-dimensional (0-D) systems can host non-trivial topology analogous to macroscopic topological materials in greater dimensions. Unlike macroscopic periodic systems with translational symmetry, zero-dimensional materials such as molecules, clusters and quantum dots can exhibit discrete rotation symmetry. The eigenstates can thus be grouped into discrete bands and Bloch-like wave functions. Since the symmetry is discrete, the Berry phase and the topological indices must be defined by discrete Wilson polygons. Here, we demonstrate non-trivial Z2 orders in two representative 0-D molecules, [m]-Cycloparaphenylene and [m]-iso-thianaphthene, where topological transitions occur when modifying the coupling between the repeating units. Similar to macroscopic topological systems in greater dimensions, localized boundary states emerge in composite nanohoops formed by segments that are topologically distinct. This opens up the possibility of non-trivial topological phases in 0-D systems.

Figures (6)

• Figure 1: The $\mathcal{Z}_2$ transitions in $\textrm{[m]-CPP}$. (a) The atomic structure of a $\textrm{[8]-CPP}$. Here, $t$ is the C-C hopping amplitude within each benzene ring, whereas $t'$ labels the hopping amplitude between neighboring units. (b) The unit cell of a PPP polymer corresponding to the case of $\textrm{[m]-CPP}$ when $m\rightarrow\infty$. (c) The mapping of MOs in [8]-CPP onto the continuous bands of a PPP at $t'=t$. The short horizontal blue lines denote the discrete energy levels of the MOs, and their corresponding mapping positions in the continuous bands are denoted by the red solid points. The gray solid lines illustrate the continuous bands of the 1-D polymer PPP. (d) The evolution of the energy levels of $\textrm{[8]-CPP}$ when modulating $t'$. Three crossings are captured. (e-g) The quantized transition of the $\mathcal{Z}_{2,N}$ index as a function of $t'$, corresponding to the three crossings shown in (d).
• Figure 2: The $\mathcal{Z}_2$ transitions in $\textrm{[m]-ITN}$. (a) An illustration of the structure of $\textrm{[8]-ITN}$, where the repeating units take a alternative structure after structure optimization. (b) The unit cell of a PITN polymer corresponding to the case of [m]ITN when $m\rightarrow\infty$. (c) The mapping of MOs in $\textrm{[8]-ITN}$ onto the continuous bands of a PITN similar to that shown in Fig.\ref{['fig:CPP']}. Here $\frac{t'}{t}=0.88$, corresponding to the crossing of the HOMO and LUMO at $\Gamma$. (d) The evolution of the energy levels of $\textrm{[8]-ITN}$ when modulating $t'$. Five crossings (i-v) are captured. (e) The $\mathcal{Z}_{2,N}$ indices as a function of $t'$, corresponding to the five crossings shown in (d). The transition in (ii) is trivial, whereas all others are topologically non-trivial. (f) The explanation of the trivial transition for Case (ii). The left and the right panels show the different band curvature before (left) and after (right) the crossing.
• Figure 3: The geometric meaning of the discrete Berry phase and the $\mathcal{Z}_2$ index. The solid red points illustrate the accumulative Berry phase when calculating the index $\mathcal{Z}_{2,N=5}$ of [6]-ITN. The magnitude of each point is normalized to the MO index when looping through the $6\times5=30$ MOs within the first five bands. The blue arcs with arrows indicate the order of Berry phase accumulation. (a) The topologically non-trivial case of $\mathcal{Z}_{2,N=5}=1$ corresponds to a net Berry phase of $\pi$. Smooth gauge transformations can deform the path into a line segment that must contain the origin. (b) The trivial case where the total Berry phase is $0$, such that the path can deform into a line segment excluding the origin.
• Figure 4: The topological boundary state in 0-D composite topological nanohoops. (a) The structure of cyclic $\textrm{[m]-ITN-[p]-CPP}$. The dashed line denotes the C-C bond connecting the first unit of the [m]-ITN segment and the last one in the [p]-CPP segment. (b) The spectrum near the HOMO-LUMO gap of $\textrm{[18]-ITN-[18]-CPP}$ when modifying the single parameter $t'/t$. (c) The average electron density resolved on the repeating units when the two segments are topologically distinct. The error bars denote the standard deviation among $200$ on-site white noise samples. (d) The electron density similar to (c) but when the two segments are topologically equivalent.
• Figure 5: DFT results and the $\mathcal{Z}_2$ transition. (a-c) Three representative DFT band structures of [8]CPP performed in ORCA. The scatters denote the discrete molecular orbital energies, whereas the solid lines illustrate the smooth band structure due to the continuous $k$. The blue and red colors denote the two energy bands corresponding to topological band gap. (d) The transition of the $\mathcal{Z}_2$ index when changing the inter-unit C-C bond length $d'$ measured in the fixed C-C bond length within the benzene ring.