Abstract
Necessary conditions for a soliton on a torus \({M = \mathbb{R}^m/\Lambda}\) to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be L 2 orthogonal to \({\mathbb{E}}\) , the space of parallel symmetric bilinear forms on TM, and, further, a certain symmetric bilinear form on \({\mathbb{E}}\) , called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice Λ, there is a baby Skyrme model which supports a soliton crystal of periodicity Λ. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies, in particular, that the celebrated Skyrme crystal of Castillejo et al., and Kugler and Shtrikman, passes both tests.
Similar content being viewed by others
References
Adam C., Sánchez-Guillén J., Wereszczyński A.: A Skyrme-type proposal for baryonic matter. Phys. Lett. B691, 105–110 (2010)
Adams R.A.: Sobolev Spaces. Academic Press, London (1975)
Baird, P., Eells, J.: A conservation law for harmonic maps. In: Geometry Symposium, Utrecht 1980 (Utrecht, 1980). Lecture Notes in Mathematics, vol. 894, pp. 1–25. Springer, Berlin (1981)
Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds, vol. 29. London Mathematical Society Monographs. New Series Oxford University Press, Oxford (2003)
Battye R.A., Sutcliffe P.M.: A Skyrme lattice with hexagonal symmetry. Phys. Lett. B416, 385–391 (1998)
Castillejo L., Jones P.S.J., Jackson A.D., Verbaarschot J.J.M., Jackson A.: Dense skyrmion systems. Nucl. Phys. A501, 801–812 (1989)
Derrick G.H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964)
Domokos S.K., Hoyos C., Sonnenschein J.: Deformation Constraints on Solitons and D-branes. J. High Energy Phys. 2013, 3 (2013)
Evans, L.C.: Weak convergence methods for nonlinear partial differential equations, vol. 74. In: CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1990)
Jaykka J., Speight M., Sutcliffe P.: Broken baby Skyrmions. Proc. R. Soc. Lond. A468, 1085–1104 (2012)
Jones H.F.: Groups, Representations and Physics. Adam Hilger, Bristol (1990)
Kapitanski, L.: On Skyrme’s model. In: Nonlinear Problems in Mathematical Physics and Related Topics, II. International Mathematical Series (New York), vol. 2, pp. 229–241. Kluwer/Plenum, New York (2002)
Karliner M., Hen I.: Rotational symmetry breaking in baby Skyrme models. In: Brown, G.E., Rho, M. (eds) The Multifaceted Skyrmion, pp. 179–213. World Scientific, Singapore (2010)
Klebanov I.: Nuclear matter in the skyrme model. Nucl. Phys. B262, 133–143 (1985)
Kugler M., Shtrikman S.: A new skyrmion crystal. Phys. Lett. B208, 491–494 (1988)
Lawden D.F.: Elliptic Functions and Applications. Springer, London (1989)
Lichnerowicz A.: Applications harmoniques et variétés kähleriennes. Symp. Math. Bologna 3, 341–402 (1970)
Lin F., Yang Y.: Existence of two-dimensional skyrmions via the concentration-compactness method. Comm. Pure Appl. Math. 57, 1332–1351 (2004)
Manton N.S.: Scaling identities for solitons beyond Derrick’s theorem. J. Math. Phys. 50, 032901 (2009)
Manton N.S., Sutcliffe P.M.: Topological Solitons. Cambridge University Press, Cambridge (2004)
Morrison T.J.: Functional Analysis. An Introduction to Banach Space Theory. Wiley, New York (2001)
Silva Lobo J., Ward R.S.: Skyrmion multi-walls. J. Phys. A42, 482001 (2009)
Speight J.M.: Compactons and semi-compactons in the extreme baby Skyrme model. J. Phys. A43, 405201 (2010)
Ward R.S.: Planar Skyrmions at high and low density. Nonlinearity 17, 1033–1040 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. T. Chruściel
Rights and permissions
About this article
Cite this article
Speight, J.M. Solitons on Tori and Soliton Crystals. Commun. Math. Phys. 332, 355–377 (2014). https://doi.org/10.1007/s00220-014-2104-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2104-z