The Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ superconducting four-gap temperature evolution: a multi-band Chebyshev-BdG approach

We generalize the Chebyshev-Bogoliubov-deGennes method to treat multi-band systems to address the temperature dependence of the superconducting (SC) gaps of iron based superconductors. Four SC gaps associated with different electron and hole pockets of optimally doped Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ were clearly identified by angle resolved photo-emission spectroscopy. The few approaches that reproduces with success this gap structure are based on strong-coupling theories and required many adjustable parameters. We show that an approach with a redistribution of electron population between the hole and electron pockets $\nu$ with evolving temperature reproduces the different coupling ratios $2\Delta^{\nu}(0)/k_{\rm B} T_c$ in these materials. We define the values that fit the four zero temperature gaps $\Delta^{\nu}(0)$ and after that all $\Delta^{\nu}(T)$ is obtained without any additional parameter.