Chambers's formula for the graphene and the Hou model with kagome periodicity and applications (Q270227)

The authors prove that, for a model considered by \textit{J.-M. Hou} [``Light-induced Hofstadter's butterfly spectrum of ultracold atoms on the two-dimensional kagome lattice'', Chinese Phys. Lett. 26, No. 12, Article ID 123701 (2009)], there exists a formula which is similar to the one obtained by \textit{W. Chambers} [``Linear network model for magnetic breakdown in two dimensions'', Phys. Rev. A 140, No. 1A, A135--A143 (1965; \url{doi:10.1103/PhysRev.140.A135})] for the Harper model. The authors establish symmetry properties of the two matrices \(J_{p,q}\) and \(K_q\). They recall that the Chambers formula plays an important role in the semi-classical analysis of the Harper model. They give an application to the case of the graphene. The authors establish the non-overlapping of the bands in the case of the kagome lattice. Also, they give as an application a semi-classical analysis near a flat band. In order to obtain the results, the authors use the pseudodifferential calculus involving the classical quantization.