Solitons on Tori and Soliton Crystals

Speight, J.M.

doi:10.1007/s00220-014-2104-z

Cited by 14 publications

(10 citation statements)

References 24 publications

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“…Indeed, symmetry arguments and the results in [14] show that both these lattice solutions are not only critical points with respect to variations of the lattice but are also stable. …”

Section: A Soliton Latticementioning

confidence: 91%

Aloof baby Skyrmions

Salmi¹,

Sutcliffe²

2014

J. Phys. A: Math. Theor.

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We show that a suitable choice for the potential term in the two-dimensional baby Skyrme model yields solitons that have a short-range repulsion and a long-range attraction. The solitons are therefore aloof, in the sense that static multi-soliton bound states have constituents that preserve their individual identities and are sufficiently far apart that tail interactions yield small binding energies. The static multi-soliton solutions are found to have a cluster structure that is reproduced by a simple binary species particle model. In the standard three-dimensional Skyrme model of nuclei, solitons are too tightly bound and are often too symmetric, due to symmetry enhancement as solitons coalesce to form bound states. The aloof baby Skyrmion results endorse a way to resolve these issues and provides motivation for a detailed study of the related three-dimensional version of the Skyrme model.

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“…Indeed, symmetry arguments and the results in [14] show that both these lattice solutions are not only critical points with respect to variations of the lattice but are also stable. …”

Section: A Soliton Latticementioning

confidence: 91%

Aloof baby Skyrmions

Salmi¹,

Sutcliffe²

2014

J. Phys. A: Math. Theor.

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“…The condition for ϕ to minimize E 2 over its SL(k, R) orbit (M = µI k ) is precisely the condition that the trace-free part of its stress tensor be L 2 orthogonal to all variations of the metric g through constant coefficient metrics [24]. Alternatively, it is the condition that ϕ should satisfy all the extended Derrick identities [19] for the energy E 0 + εE 2 + E 6 except the basic one, generated by dilations of R k .…”

Section: Remark 37mentioning

confidence: 99%

“…This has stress tensor [24] 4) where the final term denotes the (real valued) symmetric (0, 2) tensor…”

mentioning

confidence: 99%

Near BPS skyrmions and restricted harmonic maps

Speight

2015

Journal of Geometry and Physics

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Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map ϕ : (M, g) → (N, h) between Riemannian manifolds is restricted harmonic if it locally extremizes E 2 on its SDiff(M ) orbit, where SDiff(M ) denotes the group of volume preserving diffeomorphisms of (M, g), and E 2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that ϕ is restricted harmonic if and only if ϕ * h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R 3 → SU (2) and R 2 → S 2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E 2 for ϕ : R k → N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted F -criticality where F is any functional on maps (M, g) → (N, h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E 2 and E 4 , the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R 3 → S 3 ≡ SU (2) is stable restricted F -critical for every such F .

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“…Instead, they contain roots and so might not belong to the class of potentials which one wants to permit. We remark that similar BPS-type solutions on compact domains (on tori) -again leading to particular potentials -were studied in [44].…”

Section: The Holomorphic Baby Skyrme Modelmentioning

confidence: 83%

The first-order Euler-Lagrange equations and some of their uses

Adam

Santamaría

2016

J. High Energ. Phys.

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In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.PACS numbers: I. INTRODUCTIONNonlinear field theories are ubiquitous in the description of physical systems from particle physics [1] -[4] to condensed matter systems [5] -[7] and cosmology [8], where any genuine interaction is generally related to the nonlinearity of the underlying field theory. In these theories, one powerful strategy to obtain solutions of physical importance is to reduce the order of the original field equations (the Euler-Lagrange (EL) equations) of the system. The resulting equations of lower order -Bogomolnyi equations, self-duality equations, Bäcklund transformations, etc. -are easier to solve and allow to obtain a large number of relevant solutions with particular characteristics, like solitons, nonlinear waves, vortices, monopoles, instantons, etc. There exist several known methods to achieve this reduction of order, where the best-known one is probably the Bogomolnyi trick [9], [10], [11] of completing a square. To consider an example, let us assume that we have the energy functional of a field theory which for static fields may be expressed as a sum of two terms, E = d d x(A 2 + B 2 ) (typically, A depends on first derivatives, whereas B only depends on the fields). This may trivially be rewritten asIf, in addition, Q is a homotopy invariant (i.e., AB is locally a total derivative), then it does not contribute to the EL equations, and its value only depends on the boundary conditions imposed on the fields. As a consequence, E andĒ lead to the same EL equations. Further,Ē is non-negative, so E obeys the inequality E ≥ |Q| (Bogomolnyi bound) which is saturated by solutions to the reduced-order (usually, first-order) equation A = ±B (Bogomolnyi equation or BPS equation).Recently it has been observed [12] that it can be useful to partly invert the logic of this construction. That is to say, let us assume that we have two functionals (functions of the fields, their derivatives, and possibly also of the coordinates x µ ) A, B which are in some sense "duals" of each other, and which are such that the product AB is locally a total derivative (the integral Q = 2 d d xAB is a homotopy invariant). This automatically implies that the "energy functional" E = d d x(A 2 + B 2 ) is a BPS action, and the "self-duality equations" (BPS equations) A = ±B

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Solitons on Tori and Soliton Crystals

Cited by 14 publications

References 24 publications

Aloof baby Skyrmions

Aloof baby Skyrmions

Near BPS skyrmions and restricted harmonic maps

The first-order Euler-Lagrange equations and some of their uses

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