Quantum Stability for the Heisenberg Ferromagnet

TL;DR

This paper investigates the stability of spectral curves in the Heisenberg ferromagnet model, revealing that all admissible curves are stable when branch cuts are properly positioned, thus clarifying the model's spectral solution space.

Contribution

It introduces a detailed analysis of the two-cut elliptic solutions, demonstrating the stability of spectral curves in the Heisenberg ferromagnet model.

Findings

All admissible spectral curves are stable with proper branch cut placement.

The moduli space of elliptic curves exhibits a rich structure.

Insights into multi-cut solutions are provided through illustrative cases.

Abstract

Highly spinning classical strings on RxS^3 are described by the Landau-Lifshitz model or equivalently by the Heisenberg ferromagnet in the thermodynamic limit. The spectrum of this model can be given in terms of spectral curves. However, it is a priori not clear whether any given admissible spectral curve can actually be realised as a solution to the discrete Bethe equations, a property which can be referred to as stability. In order to study the issue of stability, we find and explore the general two-cut solution or elliptic curve. It turns out that the moduli space of this elliptic curve shows a surprisingly rich structure. We present the various cases with illustrations and thus gain some insight into the features of multi-cut solutions. It appears that all admissible spectral curves are indeed stable if the branch cuts are positioned in a suitable, non-trivial fashion.