布洛赫能带中的对称性与量子几何
Symmetry and Quantum Geometry in Bloch Bands
AI总结：
研究布洛赫能带中量子几何量，指出其依赖紧束缚模型参数与实空间几何有复杂性。证明若紧束缚模型与特定空间对称性兼容，使贝里曲率方差等最小化的实空间几何必服从这些对称性，且适用于多种对称系统，对霍夫施塔特模型量子几何有意义。
AI中文摘要：
近年来，量子几何量在量子系统性质的讨论中占据重要地位。其中最常讨论的量是贝里曲率的方差和量子度量张量迹的积分。尽管它们很有用，但已知它们存在一个重大复杂性：对于紧束缚模型，量子几何量不仅取决于紧束缚模型本身的参数，还取决于紧束缚模型的实空间几何，即所谓的“轨道嵌入”。因此，一个明确与几何无关的量是在所有可能的实空间几何中量子几何量的最小值。在这项工作中，我们证明，如果紧束缚模型与某些空间对称性兼容，那么使贝里曲率方差或量子度量张量迹的积分最小化的实空间几何必须服从所有这些空间对称性。我们进一步表明，该陈述适用于具有磁平移对称性和其他复合对称性的系统，这对霍夫施塔特模型的量子几何有影响。
英文摘要：
Quantum geometric quantities have featured heavily in the discussion of the properties of quantum systems in recent years. Among quantities most commonly discussed is the variance of Berry curvature and the integral of the trace of the quantum metric tensor. Despite their usefulness, it is known that they suffer from one significant complication: for a tight-binding model, the quantum geometric quantities depend not only on the parameters of the tight-binding model itself, but also the real space geometry of the tight-binding model, the so-called "orbital embedding". One explicitly geometry-independent quantity is therefore the minimal value of the quantum geometric quantity out of all possible real space geometries. In this work, we demonstrate that, if the tight-binding model is compatible with certain spatial symmetries, then the real space geometry that minimizes the variance of the Berry curvature or the integral of the trace of the quantum metric tensor must obey all those spatial symmetries. We further show that the statement is applicable to systems with magnetic translation symmetries and other composite symmetries, with implications for the quantum geometry of the Hofstadter model.
