Layer Dependence of Charge Distribution and Electronic Structure of HgBa$_2$Ca$_4$Cu$_5$O$_{12}$

Recent experimental observation of layer-dependent properties of the five-layer superconducting cuprate HgBa$_2$Ca$_4$Cu$_5$O$_{12+\delta}$ has inspired intensive interest. In general, multi-layer cuprates are of great interest because the influence of the apical oxygen $p$ states, the main source of the material dependence of cuprate structural and electronic properties [1], could be significantly layer-dependent. In this talk, the layer dependence of the charge distribution and electronic structure of HgBa$_2$Ca$_4$Cu$_5$O$_{12}$ will be investigated via the recently developed many-body Wannier-state method [1]. Possible implications on modulation of local pairing gaps, hole mobility, ``super-repulsion'' [1], and electron-phonon interaction among these distinct CuO$_2$ layers will be discussed, in connection with the intriguing experimental findings of coexisting superconducting and antiferromagnetic orders as well as strong interlayer charge inhomogeneity. Work partially supported by DOE-CMSN. \hspace {0.2cm} [1] W.-G. Yin and W. Ku, cond-mat/0702469.