Systematic enumeration of self-consistency solutions at fixed particle number

Develop a systematic procedure for finding all solutions of the four coupled self-consistency equations for the order parameters of doped, biased Bernal bilayer graphene at a specified particle number, in order to determine the ground state at fixed doping and displacement field.

Background

The mean-field theory reduces the problem to four coupled nonlinear equations for the sector-dependent order parameters. For a fixed dimensionless chemical potential, the numerical procedure enumerates combinations of roots across the monotonic branches of an auxiliary function and then compares their free energies.

The unresolved difficulty is to perform this search systematically at fixed particle number rather than fixed chemical potential. In the paper, the authors instead solve the equations for many chemical potentials and compare states whose particle numbers fall within a small interval, which provides an approximate way to identify the ground state but does not constitute a complete fixed-particle-number enumeration.