Title: Theory of ground states for classical Heisenberg spin systems IV,
Abstract: We extend the theory of ground states of classical Heisenberg spin systems
previously published to the case where the interaction with an external
magnetic field is described by a Zeeman term. The ground state problem for the
Heisenberg-Zeeman Hamiltonian can be reduced first to the relative ground state
problem, and, in a second step, to the absolute ground state problem for pure
Heisenberg Hamiltonians depending on an additional Lagrange parameter. We
distinguish between continuous and discontinuous reduction. Moreover, there are
various general statements about Heisenberg-Zeeman systems that will be proven
under most general assumptions. One topic is the connection between the minimal
energy functions $E_{min}$ for the Heisenberg energy and $H_{min}$ for the
Heisenberg-Zeeman energy which turn out to be essentially mutual
Legendre-Fenchel transforms. This generalization of the traditional Legendre
transform is especially suited to cope with situations where the function
$E_{min}$ is not convex and consequently there is a magnetization jump at a
critical field. Another topic is magnetization and the occurrence of threshold
fields $B_{thr}$ and saturation fields $B_{sat}$, where we provide a general
formula for the latter. We suggest a distinction between ferromagnetic and
anti-ferromagnetic systems based on the vanishing of $B_{sat}$ for the former
ones. Parabolic systems are defined in such a way that $E_{min}$ and $H_{min}$
have a particularly simple form and studied in detail. For a large class of
parabolic systems the relative ground states can be constructed from the
absolute ground state by means of a so-called umbrella family. Finally we
provide a counter-example of a parabolic system where this construction is not
possible.