  An interacting electron liquid in two (2D) and three (3D) dimensions may
undergo a paramagnetic-to-ferromagnetic quantum spin polarization transition at
zero applied magnetic field, driven entirely by exchange interactions, as the
system density ($n$) is decreased. This is known as Bloch ferromagnetism. We
show theoretically that the application of an external magnetic field ($B$),
which directly spin polarizes the system through Zeeman spin splitting, has an
interesting effect on Bloch ferromagnetism if the applied field and carrier
density are both decreased (from some initial applied high magnetic field at a
high carrier density) in a power-law manner, $B\sim n^p$. For $p<p_c$, with
$p_c= 1 (2/3)$ in $2(3)$D, the system remains either fully spin-polarized or
undergoes a single transition from a partially spin-polarized (with two Fermi
surfaces corresponding to spin up and down electrons) to a fully spin-polarized
state (with a single Fermi surface of one spin) as the density and field
decrease, depending on whether the starting point is partially spin-polarized
or fully spin-polarized. However, for $p>p_c$, the system may undergo two
transitions if starting from the fully spin-polarized state: first, a weak
second order transition at high density and field from the field-induced fully
polarized phase to the partially polarized phase; and then, at a lower field
and density, a reentrant first order transition back to the fully
spin-polarized phase again with a single Fermi surface.
