The recent discovery of copper-substituted lead phosphate apatite, also known as LK-99, has caught much attention owing to certain experimental evidence of room-temperature superconductivity, although this claim is currently under intensive debate. Be it superconducting or not, we show that the normal state of this material has peculiar quantum geometrical properties that may be related to the magnetism and the mechanism for flat band superconductivity. Based on a recently proposed spinless two-band tight-binding model for the Pb–Cu hexagonal lattice subset of the crystalline structure, which qualitatively captures the two flat bands in the band structure, we elaborate the highly anisotropic Berry curvature and quantum metric in the regions of Brillouin zone where one flat band is above and the other below the Fermi surface. In these regions, the Berry curvature has a pattern in the planar momentum that remains unchanged along the out-of-plane momentum. Moreover, the net orbital magnetization contributed from the Berry curvature is zero, signifying that the magnetism in this material should come from other sources. The quantum metric has a similar momentum dependence, and its two planar components are found to be roughly the same but the out-of-plane component vanishes, hinting that the superfluid stiffness of the flat band superconductivity, shall it occur, may be quite anisotropic.

中文翻译：

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铜取代磷酸铅磷灰石的贝里曲率和量子度量

最近发现的铜取代磷酸铅磷灰石（也称为 LK-99）由于室温超导性的某些实验证据而引起了广泛关注，尽管这一说法目前正在激烈争论。无论是否超导，我们都表明这种材料的正常状态具有特殊的量子几何性质，这可能与磁性和平带超导机制有关。基于最近提出的晶体结构 Pb-Cu 六方晶格子集的无自旋双能带紧束缚模型，该模型定性地捕获了能带结构中的两个平带，我们详细阐述了高度各向异性的贝里曲率和量子度量布里渊区的区域，其中一个平带位于费米表面上方，另一个位于费米表面下方。在这些区域中，贝里曲率的平面动量模式沿面外动量保持不变。此外，贝里曲率贡献的净轨道磁化强度为零，这意味着该材料中的磁性应该来自其他来源。量子度量具有类似的动量依赖性，并且发现它的两个平面分量大致相同，但平面外分量消失，这暗示平带超导性的超流体刚度（如果发生的话）可能是相当各向异性的。