Non-monotonic Fermi surface geometry and its correlation with real-space ordering in the bilayer magnetoresistive oxide La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$

Angle-resolved photoemission spectroscopy was used to study the Fermi surface and nesting effects for a wide range of doping levels (x) of La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$. While band structure calculations indicate a monotonic trend in the size of each piece of the Fermi surface, our data indicates non-monotonic trends which are consistent with the evolution of the charge modulations, unequivocally confirming the direct connection between the nesting and the charge correlation vectors. This may be classified as a key example of a system with a strongly k-dependent self-energy.