原子带排列不变的拓扑指数的无监督可解释学习
Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands
摘要：
具有内部或空间对称性的多带绝缘布洛赫哈密尔顿算子，如粒子-空穴或反转，可能具有拓扑断开的平凡原子极限（动量无关）哈密尔顿算子。我们提出了一种基于神经网络的协议，用于找到拓扑相关的指数，这些指数在这些平凡的原子极限哈密顿之间的变换下是不变的，从而对应于带绝缘体的标准分类。这项工作扩展了参考文献[1]中介绍的用于无监督学习的“拓扑数据增强”方法，同时推广和简化了数据生成方案，并引入了适合$Z_n$分类的神经网络的特殊“mod”层。通过以保持连续性的离散表示的方式使种子对象变形来生成训练数据的集合。为了将学习集中在拓扑相关的指数上，在变形过程之前，我们将种子布洛赫哈密顿算子与一组完整的尊重拓扑的平凡原子带堆叠在一起。然后将获得的数据集用于训练可解释的神经网络，该神经网络专门设计用于通过学习物理相关的动量空间量来捕获拓扑特性，即使在晶体对称类中也是如此。
英文摘要：
Multi-band insulating Bloch Hamiltonians with internal or spatial symmetries, such as particle-hole or inversion, may have topologically disconnected sectors of trivial atomic-limit (momentum-independent) Hamiltonians. We present a neural-network-based protocol for finding topologically relevant indices that are invariant under transformations between such trivial atomic-limit Hamiltonians, thus corresponding to the standard classification of band insulators. The work extends the method of "topological data augmentation" for unsupervised learning introduced in Ref. [1] by also generalizing and simplifying the data generation scheme and by introducing a special "mod" layer of the neural network appropriate for $Z_n$ classification. Ensembles of training data are generated by deforming seed objects in a way that preserves a discrete representation of continuity. In order to focus the learning on the topologically relevant indices, prior to the deformation procedure we stack the seed Bloch Hamiltonians with a complete set of symmetry-respecting trivial atomic bands. The obtained datasets are then used for training an interpretable neural network specially designed to capture the topological properties by learning physically relevant momentum space quantities, even in crystalline symmetry classes.
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