Theory of ground states for classical Heisenberg spin systems IV

TL;DR

This paper extends the theory of classical Heisenberg spin systems to include external magnetic fields, analyzing ground states, energy functions, and magnetization phenomena with new mathematical insights and specific system classifications.

Contribution

It introduces a generalized Legendre-Fenchel transform approach, analyzes ground state reductions, and distinguishes ferromagnetic and antiferromagnetic systems based on saturation fields.

Findings

Connection between $E_{min}$ and $H_{min}$ as Legendre-Fenchel transforms

General formulas for saturation and threshold fields

Construction of relative ground states from absolute ground states in parabolic systems

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.