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Stable Quasicrystalline Ground States

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Abstract

We give strong evidence that noncrystalline materials such as quasicrystals or incommensurate solids are not exceptions, but rather are generic in some regions of phase space. We show this by constructing classical lattice-gas models with translation-invariant finite-range interactions and with a unique quasiperiodic ground state which is stable against small perturbations of two-body potentials. More generally, we provide a criterion for stability of nonperiodic ground states.

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REFERENCES

  1. P. W. Anderson, Basic Notions of Condensed Matter Physics (Benjamin, Menlo Park, 1984), p. 12.

    Google Scholar 

  2. S. G. Brush, Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager (Princeton University Press, Princeton, 1983), p. 277.

    Google Scholar 

  3. B. Simon, in Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (Birkhauser-Verlag, Bassel, 1984), p. 442.

    Google Scholar 

  4. G. E. Uhlenbeck, in Fundamental Problems in Statistical Mechanics II, edited by E. G. D. Cohen (Wiley, New York, 1968), p. 16.

    Google Scholar 

  5. G. E. Uhlenbeck, in Statistical Mechanics; Foundations and Applications, edited by T. A. Bak (Benjamin, New York, 1967), p. 581.

    Google Scholar 

  6. C. Radin, Low temperature and the origin of crystalline symmetry, Int. J. Mod. Phys. B1:1157 (1987).

    Google Scholar 

  7. C. Radin, Global order from local sources, Bull. Amer. Math. Soc. 25:335 (1991).

    Google Scholar 

  8. S. Aubry, Devil 's staircase and order without periodicity in classical condensed matter, J. Physique 44:147 (1983).

    Google Scholar 

  9. Aperiodicity and Order Vol. 1 and 2, edited by M. V. Jarić (Academic Press, London, 1987, 1989).

    Google Scholar 

  10. M. Widom, K. J. Strandburg, and R. H. Swendsen, Quasicrystal ground state, Phys. Rev. Lett. 58:706 (1987).

    Google Scholar 

  11. S. Narasimhan and M. V. Jarić, Icosahedral quasiperiodic ground states? Phys. Rev. Lett. 62:454 (1989).

    Google Scholar 

  12. A. P. Smith and D. A. Rabson, Comment on “Icosahedral quasiperiodic ground states?” Phys. Rev. Lett. 63:2768 (1989).

    Google Scholar 

  13. S. Narasimhan and M. V. Jarić reply Phys. Rev. Lett. 63:2769 (1989).

    Google Scholar 

  14. Z. Olami, Stable dense icosahedral quasicrystals, Phys. Rev. Lett. 65:2559 (1990).

    Google Scholar 

  15. A. P. Smith, Stable one-component quasicrystals, Phys. Rev. B 43:11635 (1991).

    Google Scholar 

  16. C. Radin, Tiling, periodicity, and crystals, J. Math. Phys. 26:1342 (1985).

    Google Scholar 

  17. C. Radin, Crystals and quasicrystals: a lattice gas model, Phys. Lett. 114A:381 (1986).

    Google Scholar 

  18. C. Radin, Crystals and quasicrystals: a continuum model, Commun. Math. Phys. 105:385 (1986).

    Google Scholar 

  19. J. Mi\({\text{e}}\)kisz and C. Radin, The unstable chemical structure of the quasicrystalline alloys, Phys. Lett. 119A:133 (1986).

    Google Scholar 

  20. J. Mi\({\text{e}}\)kisz, Many phases in systems without periodic ground states, Commun. Math. Phys. 107:577 (1986).

  21. J. Mi\({\text{e}}\)kisz, Classical lattice gas model with a unique nondegenerate but unstable periodic ground state configuration, Commun. Math. Phys. 11:533 (1987).

  22. J. Mi\({\text{e}}\)kisz, A microscopic model with quasicrystalline properties, J. Stat. Phys. 58:1137 (1990).

  23. C. Radin, Disordered ground states of classical lattice models, Rev. Math. Phys. 3:125 (1991).

    Google Scholar 

  24. J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58:116 (1952).

    Google Scholar 

  25. Pirogov and Y. G. Sinai, Phase diagrams of classical lattice systems I, Theor. Math. Phys. 25:1185 (1975).

    Google Scholar 

  26. Pirogov and Y. G. Sinai, Phase diagrams of classical lattice systems II, Theor. Math. Phys. 26:39 (1976).

    Google Scholar 

  27. Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results (Pergamon Press, Oxford, 1982).

    Google Scholar 

  28. R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12:177 (1971).

    Google Scholar 

  29. B. Grünbaum and G. C. Shephard, Tilings and Patterns (Freeman, New York, 1986).

    Google Scholar 

  30. S. Mozes, Tilings, substitutions, and dynamical systems generated by them, J. d'Analyse Math. 53:139 (1989).

    Google Scholar 

  31. B. Tsirelson, A robust nonperiodic tiling system, preprint of Tel Aviv University, School of Mathematical Sciences (1992).

  32. W. Li, H. Park, and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66:1 (1992).

    Google Scholar 

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  1. Institute of Applied Mathematics and Mechanics, Warsaw University, 02-097, Warsaw, Poland

    Jacek Miekisz

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  1. Jacek Miekisz
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Miekisz, J. Stable Quasicrystalline Ground States. Journal of Statistical Physics 88, 691–711 (1997). https://doi.org/10.1023/B:JOSS.0000015168.25151.22

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  • DOI: https://doi.org/10.1023/B:JOSS.0000015168.25151.22

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