低维拓扑绝缘体的绝对与相对不变量
Absolute and relative invariants for low-dimensional topological insulators
AI总结：
本文针对低维拓扑绝缘体，比较了投影值映射的三种等价关系，揭示了绝对不变量（如陈数、Fu-Kane-Mele）与相对同伦不变量的出现条件，为带隙量子系统提供了统一几何框架。
AI中文摘要：
受拓扑绝缘体的启发，本文分析了在d维环面（d ≤ 2）上、受Altland-Zirnbauer-Cartan（AZC）对称性（时间反演、粒子-空穴和手性）约束的投影值映射的分类方案。主要目标是比较这些映射的三种等价概念：Murray-von Neumann等价、酉等价和同伦等价。分析依赖于在伪周期性约束下为投影值域构造对称基，其中选定的基向量在环绕环面时获得相位。若d=1，则所有AZC类均存在完全周期性的对称基。若d=2，拓扑障碍表现为非平凡相位，由Z（陈数）或Z2（Fu-Kane-Mele）不变量表征。这些不变量作为绝对不变量，对Murray-von Neumann等价和酉等价进行分类。对于同伦等价，其行为取决于环境Hilbert空间：若该空间具有非最小维度，则同伦等价直接归结为酉等价；而在最小环境空间中，会出现额外的相对同伦不变量。这些不变量的精确值在酉共轭下发生变化，仅留下映射之间的相对拓扑相位是良定义的。最终，这为低维带隙量子系统提供了一个统一的几何框架，展示了不同的等价关系如何生成特定的绝对和相对拓扑不变量。
英文摘要：
[An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on $d$-dimensional tori, $d \le 2$, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If $d=1$, fully periodic symmetric bases exist across all AZC classes. If $d=2$, topological obstructions appear as non-trivial phases characterized by $\mathbb{Z}$ (Chern number) or $\mathbb{Z}_2$ (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.
