Instability induced by exchange forces in a 2D electron gas in a magnetic field with uniform gradient

The exchange interaction is investigated theoretically for electrons confined to a 2D sample placed in a linearly varying magnetic field perpendicular to the plane. Unusual and interesting behavior is predicted: starting from zero electrons, as one adds electrons to the system the maximum distance an electron can travel transverse to the B _z = 0 line (i.e. the system's width) increases continuously but this width will subsequently begin shrinking at some critical number. However this collapse will be reversed as the number crosses another critical value, which we estimate here. For electron parameters typical for two dimensional electron gases, the instability could be observable at sufficiently low electron densities. A Hartree Fock equation is derived. We also show that in an appropriate asymptotic limit this leads to an approximately local potential. One key lesson is that the exchange interaction is large and cannot be reasonably excluded from any valid theoretical investigation.