朝向具有理想量子几何的量子态的梯度流
Gradient flow towards quantum states with ideal quantum geometry
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中文总结 AI 辅助
本文提出基于量子度量与贝里曲率平方的梯度流方法，用于格点模型量子态，经流处理的投影子可平衡几何要求，在Wilson-Dirac与Hofstadter模型中验证了其效果。
中文摘要 AI 辅助
我们提出一种针对格点模型中量子态的梯度流方法，该方法由包含量子度量与贝里曲率平方的作用量生成。这两项项分别驱动谱投影子趋近于Bogomolny饱和态和均匀贝里曲率。我们证明，由于有限维投影子存在一个不可行定理，在格点模型中通常无法同时满足这两个条件，因此该流预计会达到一个平衡两项几何要求的非平凡不动点。对于Wilson-Dirac模型，我们展示经流处理后的投影子呈现几乎均匀的贝里曲率，同时仍接近Bogomolny界。我们进一步由该经流处理的投影子构造出短程截断的扁平化哈密顿量，得到具有近平带和近均匀贝里曲率的格点模型。我们还将该方法应用于Hofstadter模型，确认了度量项和贝里曲率项在更高陈数的陈带中的作用。
英文摘要
We propose a gradient-flow method for quantum states in lattice models, generated by an action consisting of the quantum-metric and the square of the Berry curvature. These two terms drive the spectral projector toward Bogomolny saturation and uniform Berry curvature, respectively. We show that, due to a no-go theorem for finite-dimensional projectors, the two conditions cannot in general be satisfied simultaneously in lattice models. Hence the flow is expected to approach a nontrivial fixed point that balances the two geometric requirements. For the Wilson-Dirac model, we demonstrate that the flowed projector exhibits almost uniform Berry curvature while remaining close to the Bogomolny bound. We further construct a short-range truncated flattened Hamiltonian from the flowed projector and obtain a lattice model with nearly flat bands and nearly uniform Berry curvature. We also apply the method to the Hofstadter model and confirm the roles of the metric and Berry-curvature terms in a Chern band with higher Chern number.
