广义Wannier基的局域化意味着非周期绝缘体的陈数平凡

Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry

We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, call…

Wannier衰减与Thouless猜想

Wannier decay and the Thouless conjecture

本文研究了非平凡Chern类对指数衰减Wannier函数的阻碍，并在$d=2,3$维非平凡Bloch丛中构造了符合Thouless猜想的最优衰减率Wannier函数。

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。

On Parseval frames of exponentially decaying composite Wannier functions

Let $L$ be a periodic self-adjoint linear elliptic operator in $\R^n$ with coefficients periodic with respect to a lattice $\G$, e.g. Schrödinger operator $(i^{-1}\partial/\partial_x-A(x))^2+V(x)$ with periodic magnetic and electric potentials $A,V$, or a Maxwell operator $\nabla\times\varepsilon (x)^{-1}\nabla\times$ in a periodic medium. Let also $S$ be a finite part of its spectrum separated by gaps from the rest of the spectrum. We address here the question of existence of a finite set of exponentially decaying Wannier functions $w_j(x)$ such that their $\G$-shifts $w_{j,\g}(x)=w_j(x-\g)$ …

Parseval frames of exponentially localized magnetic Wannier functions

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d \le 3$, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model $m$ occupied energy bands by a real-analytic and $\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ of orthogonal projections of rank $m$. A moving orthonormal basis o…

Localised module frames and Wannier bases from groupoid Morita equivalences

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. T…

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…

格点上霍尔绝缘体的局域化二分性

Localization Dichotomy for Hall Insulators on the Lattice

本文证明了格点上非周期霍尔绝缘体的局域化二分性猜想：带隙指数局部哈密顿量的费米投影存在指数局域化万尼尔基，当且仅当其陈数为零。

托勒密泵与多滑动准周期晶格中的通用几何诱导漂移速度

Thouless pumps and universal geometry-induced drift velocity in multi-sliding quasi-periodic lattices

本文提出了一种通用框架，用于研究多滑动准周期晶格中的托勒密泵，揭示了拓扑漂移与准布里渊区几何之间的普遍关系，并通过模拟验证了其有效性。

The Localization Dichotomy for gapped periodic quantum systems

We investigate the localization properties of gapped periodic quantum systems, modeled by a periodic or covariant family of projectors, as e.g. the orthogonal projectors on the occupied orbitals at fixed crystal momentum for a gas of non-interacting electrons. We prove a general localization dichotomy for dimension $d\leq 3$: either the system is topologically trivial i.e. all the Chern numbers vanish, or any arbitrary choice of composite Wannier functions yields an infinite expectation value for the squared position operator. Equivalently, in the topologically non-trivial phase, the localizat…

半经典输运的Hodge拓扑：反常霍尔效应与非线性Berry偶极子的坐标无关几何框架

Hodge Topology of Semiclassical Transport: A Coordinate-Free Geometric Framework for the Anomalous Hall Effect and Non-Linear Berry Dipole

利用Brillouin区的Hodge-de Rham分解建立拓扑带中反常输运的坐标无关微分几何框架，通过正则化Berry曲率2-形式得到全局光滑的代理势，统一线性与非线性输运的几何描述，并证明Hodge势与最大局域Wannier函数规范的联系。

伦德 - 雷杰几何与广义 Konno - Oono 系统的可积性

Lund--Regge Geometry and Integrability of a Generalized Konno--Oono System

研究广义 Konno - Oono 系统，通过构造依赖参数的超定线性问题，建立无穷多非平凡局部守恒律证明其可积性，还分析行波解及生成的浸入\(S^3\)曲面，极限情况可得广义 Clifford 环面相关曲面。