In this paper, we derive a real-space topological invariant that involves all energy states in the system. This global invariant, denoted by Q, is always quantized to be 0 or 1, independent of symmetries. In terms of Q, we numerically investigate topological properties of the nonchiral Rice–Mele model including random onsite potentials to show that nontrivial bulk topology is sustained for weak enough disorder. In this regime, a finite spectral gap persists, and then Q is definitely identified. We also consider sublattice polarization of disorder potentials. In this case, the energy spectrum retains a gap regardless of disorder strength so that Q is unaffected by disorder. This implies that bulk topology remains intact as long as the spectral gap opens.

中文翻译：

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无手性对称性无序稻米链拓扑的实空间公式

在本文中，我们推导了涉及系统中所有能量状态的实空间拓扑不变量。这个全局不变量用Q表示，总是被量化为 0 或 1，与对称性无关。就Q而言，我们对非手性 Rice-Mele 模型的拓扑特性进行了数值研究，包括随机现场势，以表明非平凡的体拓扑对于足够弱的无序性是持续的。在这种情况下，有限的光谱间隙仍然存在，然后Q就可以确定。我们还考虑无序势的亚晶格极化。在这种情况下，无论无序强度如何，能谱都保留间隙，因此Q不受无序影响。这意味着只要光谱间隙打开，体拓扑就保持完整。