Supersymmetry (SUSY) of a Hamiltonian dictates double degeneracy between a pair of superpartners (SPs) transformed by supercharge, except at zero energy where modes remain unpaired in many cases. Here we explore a SUSY of complete isospectrum between SPs—with paired zero modes—realized by 2D electrons in zero-flux periodic gauge fields, which can describe twisted or periodically strained 2D materials. We find their low-energy sector containing zero (or threshold) modes must be topologically non-trivial, by proving that Chern numbers of the two SPs have a finite difference dictated by the number of zero modes and energy dispersion in their vicinity. In 30° twisted bilayer (double bilayer) transition metal dichalcogenides subject to periodic strain, we find one SP is topologically trivial in its lowest miniband, while the twin SP of identical dispersion has a Chern number of 1 (2), in stark contrast to time-reversal partners that have to be simultaneously trivial or nontrivial. For systems whose physical Hamiltonian corresponds to the square root of a SUSY Hamiltonian, such as twisted or strained bilayer graphene, we reveal that topological properties of the two SUSY SPs are transferred respectively to the conduction and valence bands, including the contrasted topology in the low-energy sector and identical topology in the high-energy sector. This offers a unified perspective for understanding topological properties in many flat-band systems described by such square-root models. Both types of SUSY systems provide unique opportunities for exploring correlated and topological phases of matter.

中文翻译：

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超对称决定了周期性规范场中的拓扑以及应变和扭曲二维材料中的实现


哈密​​顿量的超对称性 (SUSY) 决定了通过增压变换的一对超伙伴 (SP) 之间的双重简并性，但在零能量下，模式在许多情况下保持不配对。在这里，我们探索了 SP 之间完整等谱的 SUSY（具有成对的零模式），由零通量周期性规范场中的二维电子实现，它可以描述扭曲或周期性应变的二维材料。通过证明两个 SP 的陈数具有由零模数和其附近的能量色散决定的有限差，我们发现它们包含零（或阈值）模的低能量扇区在拓扑上一定是不平凡的。在受到周期性应变的 30° 扭曲双层（双双层）过渡金属二硫化物中，我们发现一个 SP 在其最低微带中是拓扑平凡的，而相同色散的孪生 SP 的陈数为 1 (2)，这与时间逆转伙伴必须同时平凡或不平凡。对于物理哈密顿量对应于 SUSY 哈密顿量的平方根的系统，例如扭曲或应变双层石墨烯，我们揭示了两个 SUSY SP 的拓扑性质分别转移到导带和价带，包括低层中的对比拓扑-能源部门和高能部门的相同拓扑。这为理解由此类平方根模型描述的许多平带系统中的拓扑特性提供了统一的视角。这两种类型的 SUSY 系统都为探索物质的相关相和拓扑相提供了独特的机会。