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Register a new userClassically Isospinning Hopf Solitons
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We perform full three-dimensional numerical relaxations of isospinning Hopf solitons with Hopf charge up to 8 in the Skyrme-Faddeev model with mass terms included. We explicitly allow the soliton solution to deform and to break the symmetries of the static configuration. It turns out that the model with its rich spectrum of soliton solutions, often of similar energy, allows for transmutations, formation of new solution types and the rearrangement of the spectrum of minimal-energy solitons in a given topological sector when isospin is added. We observe that the shape of isospinning Hopf solitons can differ qualitatively from that of the static solution. In particular the solution type of the lowest energy soliton can change. Our numerical results are of relevance for the quantization of the classical soliton solutions.
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We investigate how isospin affects the geometrical shape and energy of classical soliton solutions of topological charges $B=1-4,8$ in the Skyrme model. The novel approach in our work is that we study classically isospinning Skyrmions beyond the rigid-body approximation; that is, we explicitly allow the soliton solutions to deform and to break the symmetries of the static configurations. Our fully three-dimensional relaxation calculations reveal that the symmetries of isospinning Skyrme solitons can differ significantly from the ones of the static configurations. In particular, isospinning Skyrmion solutions can break up into lower-charge Skyrmions, can deform into new solution types that do not exist at vanishing angular frequency $omega$ or energy degeneracy can be removed. These types of deformations have been largely ignored in previous work on modeling nuclei by quantized Skyrmion solutions.
The Nicole model is a conformal field theory in three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to numerically construct soliton solutions for all Hopf charges from one to eight. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than two and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme-Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.
The Aratyn-Ferreira-Zimerman (AFZ) model is a conformal field theory in three-dimensional space. It has solutions that are topological solitons classified by an integer-valued Hopf index. There exist infinitely many axial solutions which have been found analytically. Axial, knot and linked solitons are found numerically to be static solutions using a modified volume preserving flow for Hopf index one to eight, allowing for comparison with other Hopf soliton models. Solutions include a static trefoil knot at Hopf index five. A one-parameter family of conformal Skyrme-Faddeev (CSF) models, consisting of linear combinations of the Nicole and AFZ models, are also investigated numerically. The transition of solutions for Hopf index four is mapped across these models. A topological change between linked and axial solutions occurs, with fewer models permitting axial solitons than linked solitons at Hopf index four.
The problem of constructing internally rotating solitons of fixed angular frequency $omega$ in the Faddeev-Skyrme model is reformulated as a variational problem for an energy-like functional, called pseudoenergy, which depends parametrically on $omega$. This problem is solved numerically using a gradient descent method, without imposing any spatial symmetries on the solitons, and the dependence of the solitons energy on $omega$, and on their conserved total isospin $J$, studied. It is found that, generically, the shape of a soliton is independent of $omega$, and that its size grows monotonically with $omega$. A simple elastic rod model of time-dependent hopfions is developed which, despite having only one free parameter, accounts well for most of the numerical results.
In the Skyrme model atomic nuclei are modelled as quantized soliton solutions in a nonlinear field theory of pions. The mass number is given by the conserved topological charge $B$ of the solitons. Conventionally, Skyrmions are semiclassically quantized within the rigid body approach. In this approach Skyrmions are effectively treated as rigid rotors in space and isospace that is it is assumed that Skyrmions do not deform at all when they spin and isospin. This approximation resulted in qualitative and encouraging quantitative agreement with experimental nuclear physics data. In this talk, we point out that the theoretical agreement could be further improved by allowing classical Skyrmion solutions to deform as they spin and isospin. As a first step towards a better understanding of how nuclei can be approximated by classically spinning and isospinning soliton solutions, we study how classical Skyrmion solutions of topological charges $B=1-4,8$ deform when classical isospin is added.
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