Hopf Solitons on the Lattice

TL;DR

This paper explores lattice analogs of Hopf solitons in the Skyrme-Faddeev model, demonstrating their existence, similarity to continuum solitons, and a specific energy-topology relation on a cubic lattice.

Contribution

It introduces the concept of Hopf solitons on a lattice, showing their existence and properties analogous to continuum models, which was not previously established.

Findings

Lattice Hopf solitons exist for suitable interactions.

They exhibit similar structure to continuum solitons.

Energy scales with Hopf number as E ≈ c H^{3/4}.

Abstract

Hopf solitons in the Skyrme-Faddeev model -- S^2-valued fields on R^3 with Skyrme dynamics -- are string-like topological solitons. In this Letter, we investigate the analogous lattice objects, for S^2-valued fields on the cubic lattice Z^3 with a nearest-neighbour interaction. For suitable choices of the interaction, topological solitons exist on the lattice. Their appearance is remarkably similar to that of their continuum counterparts, and they exhibit the same power-law relation E \approx c H^{3/4} between the energy E and the Hopf number H.