3D van der Waals $\sigma$-model and its Topological Excitations

TL;DR

This paper explores a 3D van der Waals nonlinear sigma model on a sphere, revealing two types of topological excitations similar to vortices and instantons, and discusses potential topological phase transitions.

Contribution

It introduces the existence of two distinct topological excitations in the 3D van der Waals sigma model and analyzes their energies and interactions.

Findings

Hedgehog excitations have logarithmic energies.

Two types of topological excitations are identified: vortices and instantons.

Potential for topological phase transitions is discussed.

Abstract

It is shown that 3D vector van der Waals (conformal) nonlinear $\sigma$-model (NSM) on a sphere $S^2$ has two types of topological excitations reminiscent vortices and instantons of 2D NSM. The first, the hedgehogs, are described by homotopic group $\pi_2(S^2) = \mathbb {Z}$ and have the logarithmic energies. They are an analog of 2D vortices. The energy and interaction of these excitations are found. The second, corresponding to 2D instantons, are described by hpmotopic group $\pi_3(S^2) = \mathbb {Z}$ or the Hopf invariant $H \in \mathbb {Z}$. A possibility of the topological phase transition in this model and its applications are briefly discussed.