AI 中文总结
研究量子几何张量计算中因晶胞惯例选择产生的问题，引入基于协变导数的物理量子几何张量解决，揭示标准 $k\cdot p$ 理论不足，提出几何工程可独立调节几何响应。
AI 中文摘要
量子几何张量包含量子度量和贝里曲率，是现代凝聚态物理的核心概念。然而，通过布洛赫投影算符的 $k$ 导数进行的标准计算，在晶胞惯例选择上存在基本模糊性，特别是在处理晶胞内轨道位置时。我们通过引入一个基于协变导数定义的与惯例无关的物理量子几何张量来解决这一不一致性，该协变导数明确纳入了完整的位置算符。我们证明这种表述由通过派尔斯代换对物理电流的微观推导唯一确定。值得注意的是，我们发现对于具有键序能隙的系统，标准 $k\cdot p$ 有效理论存在一阶失效，这表明在应用时需谨慎。最后，我们提出几何工程作为一种新的设计范式，能够在不改变能量色散的情况下独立调节几何响应。
英文摘要
The Quantum Geometric Tensor, encompassing the quantum metric and Berry curvature, is a central concept in modern condensed matter physics. However, its standard calculation via $k$-derivatives of the Bloch projector conceals a fundamental ambiguity regarding the choice of unit-cell convention, specifically in the treatment of intra-cell orbital positions (i.e., with or without the orbital position $e^{ikx_α}$). We resolve this inconsistency by introducing a convention-independent physical QGT defined via a covariant derivative that explicitly incorporates the full position operator. We demonstrate that this formulation is uniquely mandated by the microscopic derivation of the physical current via the Peierls substitution. Notably, we uncover a leading-order failure in standard $k \cdot p$ effective theories for systems with bond-ordered gaps, identifying a need for caution in their application. Finally, we propose geometric engineering as a new design paradigm, enabling the independent tuning of geometric responses without altering the energy dispersion.
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