Introduction to dimensional reduction of fermions

We present a comprehensive pedagogical introduction to the dimensional reduction protocol (DRP), a versatile framework for analyzing instabilities and critical points in interacting fermionic systems. The DRP simplifies the study of many-body problems by systematically reducing their effective spatial dimension while retaining essential physics. This method works for electron gases in a diverse array of settings: in any number of spatial dimensions, in the presence of Zeeman fields, with spin-orbit coupling, including repulsive or attractive interactions. Focusing on two-point correlation functions, the DRP identifies a minimal subspace relevant for capturing analytic properties, facilitating efficient computation of critical phenomena in electronic systems. This work outlines the assumptions, proof, and applications of the DRP, emphasizing its simplicity and broad applicability for future studies in correlated electron physics.