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Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions

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Abstract

We study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible connection with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino, focusing close to the helimagnet/ferromagnet transition point corresponding to the critical value of the frustration parameter \(\alpha=4\). We reformulate this problem for any \(\alpha\geq0\) in the framework of surface energies in nonconvex discrete systems with nearest neighbours ferromagnetic and next-to-nearest neighbours antiferromagnetic interactions and we link it to the gradient theory of phase transitions, by showing a uniform equivalence by \(\varGamma \)-convergence on \([0,4]\) with Modica-Mortola type functionals.

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Acknowledgements

We are grateful to Andrea Braides for suggesting this problem, and we would like to thank him for his advices and many fruitful discussions. We also thank Marco Cicalese, Francesco Solombrino and Leonard Kreutz for some interesting remarks leading to improve the manuscript. The first author gratefully acknowledges the hospitality of the Department of Mathematics, University of Rome “Tor Vergata”, where a substantial part of this work has been carried out, and the financial support of PRIN 2010, project “Discrete and continuum variational methods for solid and liquid crystals”.

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Authors and Affiliations

  1. Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Giovanni Scilla

  2. Department of Mathematics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Valerio Vallocchia

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  1. Giovanni Scilla
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  2. Valerio Vallocchia
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Correspondence to Giovanni Scilla.

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Scilla, G., Vallocchia, V. Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions. J Elast 132, 271–293 (2018). https://doi.org/10.1007/s10659-017-9668-8

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  • DOI: https://doi.org/10.1007/s10659-017-9668-8

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