发表机构
University of Zurich; Kyoto University; ETH Zurich; University of Southern Denmark（苏黎世大学; 京都大学; 苏黎世联邦理工学院; 南丹麦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究 AZ 对称类中多重能带简并的拓扑表征，通过分析其稳定性源于动量空间局部 AZ 对称性，利用包围球上节点流形的特性及能带不变量建立双向对应关系，为多能带模型多重简并表征提供基础。
AI 中文摘要
传统上，拓扑能带简并由能谱保持间隙的包围球上定义的不变量来表征。对于所有十个阿尔特兰 - 齐恩鲍尔（AZ）对称类中的最小简并情况，此计划已完成，而高阶简并几乎仅在晶体对称保护下进行研究。在这项工作中，我们表征了一般的 n 重能带简并，其稳定性仅源于在动量空间中局部作用的 AZ 对称性。我们发现它们的余维数随 n 二次增长，将此类多重节点置于结合物理动量与调谐参数或合成维度的参数空间中。然而，包围球范式面临一个基本障碍：在 n 重能带节点处相交的两个（n - 1）重简并轨迹会穿透每个包围球，这意味着不存在统一的能谱间隙（因此也不存在标准的同伦分类）。在此，我们将此障碍转化为诊断本身。在两个轨迹与包围球相交的节点流形上，互补能谱间隙得以恢复，这使我们能够用传统能带不变量对每个进行表征。这使我们能够建立双向对应关系：（1）当节点流形在包围球上稳健链接时，多重节点受到拓扑保护；（2）一个节点流形循环上的能带不变量编码了它们与另一个节点流形循环的链接数。我们对所有十个 AZ 类的最小模型进行了此表征，在有显式参数化的地方计算能带不变量。我们的结果将多重能带拓扑重塑为动量空间中链接节点流形的拓扑，而我们的方法为具有任意多个能带的模型中多重能带简并的一般表征提供了基础。
英文摘要
Topological degeneracies of energy bands in crystalline matter are conventionally characterized by invariants computed on an enclosing sphere over which the spectrum remains gapped, a program completed for minimal degeneracies in all ten Altland-Zirnbauer (AZ) symmetry classes. Higher-order degeneracies have instead been studied almost exclusively under crystalline-symmetry protection. Here, we characterize generic $n$-fold degeneracies stabilized solely by AZ symmetries acting locally in momentum space. Their codimension grows quadratically with $n$, placing multifold nodes in parameter spaces combining momenta with tuning parameters or synthetic dimensions. Crucially, the enclosing-sphere paradigm faces a fundamental obstruction: two $(n\,{-}\,1)$-fold degeneracy loci emanating from the $n$-fold node necessarily pierce every choice of enclosing sphere, leaving no uniform spectral gap and thus no standard homotopy classification. We elevate this obstruction into the diagnostic itself. Namely, on the two nodal manifolds where the loci cross the sphere, complementary spectral gaps are restored, admitting conventional band invariants (Chern numbers, Stiefel-Whitney classes, and winding numbers). This observation establishes a general two-way correspondence: (1) the multifold node is topologically protected whenever the associated nodal manifolds are robustly linked, and (2) invariants on cycles of one manifold encode their linking numbers with cycles of the other. Carrying out this program for minimal models of all ten AZ classes, we recast multifold band topology as the topology of linked nodal manifolds and lay the foundation for characterizing multifold nodes in models with arbitrarily many bands.
Comments51 pages, 9 figures, 5 tables
