Berry curvature and quantum metric in copper-substituted lead phosphate apatite.

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.

A Real-Space Study of Flat Bands in Nanowires.

The flat electronic band has remarkable relevance in the strongly correlated phenomena mainly due to its reduced kinetic energy in comparison to the many-body potential energy. The formation of such bands in cubically structured nanowires is addressed in this article by means of a new independent channel method and a generalized convolution theorem developed for the Green's function including the first, second, and third neighbor interactions. A real-space renormalization method is further applied to address macroscopic-length aperiodic nanowires. We also determined the appearance condition of these flat bands, as well as their degeneracy and robustness in the face of perturbations, such as structural dislocations. Finally, the possible experimental detection of this flat band via the electronic specific heat is analyzed.