Title: Constraints on real space representations of Chern bands Show Chinese title

Title: 陈绝缘体带的真实空间表示的约束条件

Abstract: A Chern band is characterized by a Wannier obstruction indicating the absence of a basis of complete, orthogonal, and exponentially-localized states. Here, we study the properties of real space bases of a Chern band obtained by relaxing either exponential localization or orthogonality and completeness. This yields two distinct real space representations of a band with Chern number $C$: (i) a basis of complete orthogonal Wannier states which decay as power-law and (ii) a basis of exponentially-localized overcomplete non-orthogonal coherent states. For (i), we show that the power-law tail only depends on the Chern number and provide an explicit gauge choice leading to the universal asymptotic $w({\boldsymbol r}) \approx \frac{C e^{-i C \varphi_{\boldsymbol r}}}{2\pi |{\boldsymbol r}|^2}$ up to a normalized Bloch-periodic spinor. For (ii), we prove a rigorous lower bound on the spatial spread that can always be saturated for ideal bands. We provide an explicit construction of the maximally localized coherent state by mapping the problem to a dual Landau level problem where the Berry curvature and trace of the quantum metric take the roles of an effective magnetic field and scalar potential, respectively. Our coherent state result rigorously bounds the spatial spread of any localized state constructed as a linear superposition of wavefunctions within the Chern band. Remarkably, we find that such bound does not generically scale with the Chern number and provide an explicit example of an exponentially localized state in a Chern $C$ band whose size does not increase with $|C|$. Our results show that band topology can be encoded in a real space description and set the stage for a systematic study of interaction effects in topological bands in real space. Show Chinese abstract

Abstract: 一个陈带（Chern band）由表征其不存在完全、正交且指数局域化态系的Wannier阻抗（Wannier obstruction）所刻画。 在这里，我们研究通过放松指数局域化或者正交性与完备性而获得的陈带实空间基底的性质。 这导致了具有陈数 $C$ 的能带的两种不同的实空间表示：(i) 完全正交且以幂律衰减的Wannier态的基底；(ii) 指数局域化、过完备且非正交的相干态的基底。 对于 (i)，我们证明幂律尾部仅依赖于陈数，并提供一种规范选择，使得归一化的Bloch周期自旋波函数的渐近形式为 $w({\boldsymbol r}) \approx \frac{C e^{-i C \varphi_{\boldsymbol r}}}{2\pi |{\boldsymbol r}|^2}$。 对于 (ii)，我们证明了一个关于空间展布的严格下界，该下界总是可以在理想带中达到饱和。 我们通过将问题映射到一个对偶朗道能级问题来显式构造最大局域化的相干态，在此问题中，Berry曲率和量子度规的迹分别扮演有效磁场和标量势的角色。 我们的相干态结果严格限制了任何作为陈带内波函数线性叠加构造的局域态的空间展布。 令人惊讶的是，我们发现这种限制通常并不随陈数变化，并提供了陈 $C$ 带中一个指数局域化态的明确例子，其大小不会随着 $|C|$ 增大。 我们的结果表明，带拓扑可以通过实空间描述编码，并为系统研究拓扑带在实空间中的相互作用效应奠定了基础。