Triviality of Bloch and Bloch-Dirac bundles

Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII

We provide a classification of type AII topological quantum systems in dimension d=1,2,3,4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies "Quaternionic" vector bundles (a.k.a. "symplectic" vector bundles) in dimension d<4. This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d=4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we pr…

On Complex Supermanifolds with Trivial Canonical Bundle

We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods include the use of complex integral forms and the recently developed theory of superholonomy.

Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; to a large class of symmetries for a non-magnetic system, one can associate "magnetic" symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bund…

Bloch能带拓扑的Cauchy数据方法

Topology of Bloch Bands from Cauchy Data

通过Cauchy数据的投影空间和反演对称性，将Bloch能带的拓扑用极点-零点不变量刻画，并与Berry-Zak相位、Real线丛和局部系数系统建立几何联系。

坚实基础上的脆弱拓扑：一个数学视角

Fragile topology on solid grounds: a mathematical perspective

本文从数学角度研究凝聚态物理中的脆弱拓扑，证明非平凡欧拉类是构造对称相容指数局域化万尼尔函数的阻碍，且添加对称布洛赫丛可消除该阻碍，解释了拓扑脆弱性的数学机制。

Topology vs localization in synthetic dimensions

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g. in optical lattices of ultracold atoms, we discuss here $d$-dimensional periodic, gapped quantum systems for $d \le 4$, with focus on the topology of the occupied energy states. We perform this analysis by asking whether the spectral subspace below the gap can be spanned by smooth and periodic Bloch functions, corresponding to localized Wannier functions in position space. By constructing these Bloch functions inductively in the dimension, we show that if they are required to be orthonormal then in general the…

Hyperbolic band theory through Higgs bundles

Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's theorem have revealed an intrinsic role for algebro-geometric moduli spaces, notably those of stable bundles on a curve. We expand this picture to include Higgs bundles, which enjoy natural interpretations in the co…

三维情形下的最优Wannier衰减

On optimal Wannier decay in dimension three

针对三维环面陈非平凡布洛赫丛，构造最优逐点衰减$\boldsymbol{O}(|x|^{-5/2})$的标准正交Wannier基，除一个生成元外其余指数衰减，解决了arXiv:2505.01999遗留问题。