Electronic Journal of Probability

Spontaneous breaking of rotational symmetry in the presence of defects

Markus Heydenreich, Franz Merkl, and Silke Rolles

Full-text: Open access

Abstract

We prove a strong form of spontaneous breaking of rotational symmetry for a simple model of two-dimensional crystals with random defects in thermal equilibrium at low temperature. The defects consist of isolated missing atoms.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 111, 17 pp.

Dates
Accepted: 11 December 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065753

Digital Object Identifier
doi:10.1214/EJP.v19-2971

Mathematical Reviews number (MathSciNet)
MR3296527

Zentralblatt MATH identifier
1307.60142

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B21: Continuum models (systems of particles, etc.)

Keywords
Spontaneous symmetry breaking localized defects rigidity estimate

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Heydenreich, Markus; Merkl, Franz; Rolles, Silke. Spontaneous breaking of rotational symmetry in the presence of defects. Electron. J. Probab. 19 (2014), paper no. 111, 17 pp. doi:10.1214/EJP.v19-2971. https://projecteuclid.org/euclid.ejp/1465065753


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References

  • Aizenman, Michael; Jansen, Sabine; Jung, Paul. Symmetry breaking in quasi-1D Coulomb systems. Ann. Henri Poincare 11 (2010), no. 8, 1453-1485.
  • Aumann, Simon. Spontaneous breaking of rotational symmetry with arbitrary defects and a rigidity estimate, Preprint arXiv:1408.5375 [math.PR], 2014.
  • Flatley, Lisa and Theil, Florian. Face-centered cubic crystallization of atomistic configurations, Preprint arXiv:1407.0692 [math.AP], 2014.
  • Friesecke, Gero; James, Richard D.; Müller, Stefan. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002), no. 11, 1461-1506.
  • Gaal, Alexisz Tamas. Long-range order in a hard disk model in statistical mechanics. Electron. Commun. Probab. 19 (2014), no. 9, 9 pp.
  • Georgii, Hans-Otto. Gibbs measures and phase transitions. Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 2011. xiv+545 pp. ISBN: 978-3-11-025029-9
  • Le Bris, Claude; Lions, Pierre-Louis. From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 3, 291–363 (electronic).
  • Merkl, Franz; Rolles, Silke W. W. Spontaneous breaking of continuous rotational symmetry in two dimensions. Electron. J. Probab. 14 (2009), no. 57, 1705–1726.
  • International Union of Crystallography, Online Dictionary of Crystallography, http://reference.iucr.org/dictionary/Crystal, October 2014.
  • Richthammer, Thomas. Translation-invariance of two-dimensional Gibbsian point processes. Comm. Math. Phys. 274 (2007), no. 1, 81–122.
  • Theil, Florian. A proof of crystallization in two dimensions. Comm. Math. Phys. 262 (2006), no. 1, 209–236.