Coexistsing Spin and Charge Density orders in a bi-layer model of $La_3Ni_2O_7$

Using the functional Renormalization Group (fRG), we study the four-band bi-layer Hubbard model of the recently discovered diouble-layer Perovskite superconductor $La_3Ni_2)_7$ (LNO) under pressure. We focus our study on the spine and charge response in the system as the possible parent state for the superconducting response in the system. We consider two orbitals for the LNO (the $d_{d_{z^2-r^2}$ and $d_{x^2-y^2}$) orbitals and study the competition between interorbtial hybridization and interlayer hopping. Our preliminary results for hte interorbital hybridization show a strong charge and spin response at multiple momentum points. We study the impact of varying the interlayer coupling on the charge and spin fluctuations present in the system. We also consider the competition between spin and charge fluctuations on the superconducting response of hte system.