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Crystalline Motion of Interfaces Between Patterns

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Abstract

We consider the dynamical problem of an antiferromagnetic spin system on a two-dimensional square lattice \(\varepsilon \mathbb {Z}^2\) with nearest-neighbour and next-to-nearest neighbour interactions. The key features of the model include the interaction between spatial scale \(\varepsilon \) and time scale \(\tau \), and the incorporation of interfacial boundaries separating regions with microstructures. By employing a discrete-time variational scheme, a limit continuous-time evolution is obtained for a crystal in \(\mathbb {R}^2\) which evolves according to some motion by crystalline curvatures. In the case of anti-phase boundaries between striped patterns, a striking phenomenon is the appearance of some “non-local” curvature dependence velocity law reflecting the creation of some defect structure on the interface at the discrete level.

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Notes

  1. A bisectric direction has slope \(\pm 1\). A bisectric segment is a segment along the bisectric direction.

  2. Note that there is a difference in the definition of the number of layers moving in for the bisectric segments between the \(e_1\)-\(e_2\)-interface in Sect. 4 and the current \(e_1\)-\(-e_1\)-interface. In the former case, the number is measured along the normal bisectric direction while here it is measured along the horizontal direction. This is simply for arithmetic convenience.

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Acknowledgments

The authors would like to thank the hospitality of the Institute for Mathematics and Its Applications (IMA), Minnesota, it was where this project started. The second author was supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. The third author also appreciates the hosting by the Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata of his several visits which facilitated the completion of this project.

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Correspondence to Nung Kwan Yip.

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Braides, A., Cicalese, M. & Yip, N.K. Crystalline Motion of Interfaces Between Patterns. J Stat Phys 165, 274–319 (2016). https://doi.org/10.1007/s10955-016-1609-6

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