AI 中文总结
该研究针对准周期系统因无平移对称性导致传统能带理论失效的问题，开发了高效谱函数方法并引入双面神量子数，建立了准周期系统的统一谱理论，实现了类似周期系统的量子态处理能力。
AI 中文摘要
准周期系统不存在平移对称性，这使得传统能带理论失效，构成该领域的核心挑战。基于无公度能带（IEB）概念，我们通过引入两项关键进展，为准周期系统建立了统一的谱理论。其一，我们开发了一种高效的谱函数方法，利用小型截断哈密顿矩阵计算$A(k,\omega)$，无需完整对角化；该方法通过能量矩的独特相继锁定实现收敛，可获得热力学极限下的精确结果，无需有限尺寸标度。其二，我们提出准周期本征态具有双面神量子数：单个本征态在局域化相变过程中被连续追踪，携带动量与实空间的双重标记，在无公度极限下自然退化为熟悉的布洛赫动量与能带指数。结合IEB，这些进展构成了准周期系统的“能带理论”，使我们能如同在周期系统中一样便捷地定义、计算和标记量子态。
英文摘要
The absence of translational symmetry in quasiperiodic systems invalidates conventional band theory, posing the central challenge in the field. Building upon the incommensurate energy band (IEB) concept, we establish a unified spectral theory for quasiperiodic systems by introducing two key advances. First, we develop an efficient spectral function method that calculates $A(k,ω)$ using a small truncated Hamiltonian matrix, bypassing full diagonalization. It converges via a distinctive successive locking of energy moments, yielding exact thermodynamic-limit results without finite-size scaling. Second, we introduce that quasiperiodic eigenstates possess Janus quantum numbers: a single eigenstate, continuously tracked across localization transitions, carries dual labels in momentum and real space, which naturally reduce to the familiar Bloch momentum and band index in the commensurate limit. Together with IEB, these advances constitute a ``band theory'' for quasiperiodic systems, enabling us to define, compute, and label states with the same facility as in periodic ones.
