Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 073, 26 pages      arXiv:2305.18126      https://doi.org/10.3842/SIGMA.2023.073
Contribution to the Special Issue on Topological Solitons as Particles

Nudged Elastic Bands and Lightly Bound Skyrmions

James Martin Speight a and Thomas Winyard b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK

Received May 30, 2023, in final form September 25, 2023; Published online October 11, 2023

Abstract
It has become clear in recent years that the configuration space of the nuclear Skyrme model has, in each topological class, many almost degenerate local energy minima and that the number of such minima grows with the degree (or baryon number) $B$. Rigid body quantization, in which one quantizes motion on the spin-isospin orbit of just one minimum, is thus an ill-justified approximation. Instead, one should identify a (finite-dimensional) moduli space of configurations containing all local minima (for a given $B$) as well as fields interpolating smoothly between them. This paper proposes a systematic computational scheme for generating such a moduli space: one constructs an energy minimizing path between each pair of local minima, then defines the moduli space to be the union of spin-isospin orbits of points on the union of these curves, a principal bundle over a graph. The energy minimizing curves may be constructed in practice using the nudged elastic band method, a standard tool in mathematical chemistry for analyzing reaction paths and computing activation energies. To illustrate, we apply this scheme to the lightly bound Skyrme model in the point particle approximation, constructing the graphs for $5\leq B\leq 10$. We go on to complete the quantization for $B=7$, in which the graph has two vertices and a single edge. The low-lying quantum states with isospin $1/2$ do not strongly localize around either of the local energy minima (the vertices). Their energies rise monotonically with spin, conflicting with experimental data for Lithium-7.

Key words: nuclear Skyrme model; energy minimizing paths; saddle points; semi-classical quantization.

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