Journal of Statistical Physics

, Volume 165, Issue 2, pp 274–319 | Cite as

Crystalline Motion of Interfaces Between Patterns

  • Andrea Braides
  • Marco Cicalese
  • Nung Kwan YipEmail author


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.


Antiferromagnetic spin system Anti-phase boundaries Microstructures Defects Interface motion Crystalline curvature motion 



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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversity of Rome Tor VergataRomeItaly
  2. 2.Zentrum MathematikTUMünichGermany
  3. 3.Department of MathemticsPurdue UniversityWest LafayetteUSA

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