Abstract
A classical lattice-gas model is called frustrated if not all of its interactions can attain their minima simultaneously. The antiferromagnetic Ising model on the triangular lattice is a standard example.(1, 29) However, in all such models known so far, one could always find nonfrustrated interactions having the same ground-state configurations. Here we constructed a family of classical lattice-gas models with finite-range, translation-invariant, frustrated interactions and with unique ground-state measures which are not unique ground-state measures of any finite-range, translation-invariant, nonfrustrated interactions.
Our ground-state configurations are two-dimensional analogs of one-dimensional, “most homogeneous,”(13) nonperiodic ground-state configurations of infinite-range, convex, repulsive interactions in models with devil's staircases.
Our models are microscopic (toy) models of quasicrystals which cannot be stabilized by matching rules alone; competing interactions are necessary.
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REFERENCES
G. Toulouse, Commun. Phys. 2:115 (1977).
M. Born, On the stability of crystal lattices, Proc. Cambridge Phil. Soc. 36:160 (1940).
G. E. Uhlenbeck, in Statistical Mechanics; Foundations and Applications, T. A. Bak, ed. (Benjamin, New York, 1967), p. 581.
G. E. Uhlenbeck, in Fundamental Problems in Statistical Mechanics II, E. G. D. Cohen, ed. (Wiley, New York, 1968), p. 16.
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.
P. W. Anderson, Basic Notions of Condensed Matter Physics (Benjamin, Menlo Park, 1984), p. 12.
B. Simon, in Perspectives in Mathematics: Anniversary of Oberwolfach 1984 (Birkhäuser-Verlag, Bassel, 1984), p. 442.
C. Radin, Low temperature and the origin of crystalline symmetry, Int. J. Mod. Phys. B1:1157 (1987).
D. Schechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53:1951 (1984).
F. Axel and D. Gratias, eds., Beyond Quasicrystals (Les Houches, 1994) (Springer, Berlin, 1994).
C. Radin, Global order from local sources, Bull. Amer. Math. Soc. 25:335 (1991).
A. C. D. van Enter, R. Fernandez, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Stat. Phys. 72:879 (1993).
J. Hubbard, Generalized Wigner lattices in one-dimension and some applications to tetracyanoquinodimethane (TCNQ) salts, Phys. Rev. 17:494 (1978).
P. Bak and R. Bruinsma, One-dimensional Ising model and the complete devil's staircase, Phys. Rev. Lett. 49:249 (1982).
S. Aubry, Exact models with a complete devil's staircase, J. Phys. C16:2497 (1983).
R. B. Griffiths, Frenkel-Kontorova models of commensurate-incommensurate phase transitions, in Fundamental Problems in Statistical Mechanics VII, H. van Beijeren, ed. (Elsevier, Amsterdam, 1990).
R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12:177 (1971).
S. Mozes, Tilings, substitutions, and dynamical systems generated by them, J. d'Analyse Math. 53:139 (1989).
B. Grünbaum and G. C. Shephard, Tilings and Patterns (Freeman, New York, 1986).
M. Senechal, Quasicrystals and Geometry (Cambridge University Press, 1995).
C. Radin, Tiling, periodicity, and crystals, J. Math. Phys. 26:1342 (1985).
C. Radin, Crystals and quasicrystals: a lattice gas model, Phys. Lett. 114A:381 (1986).
J. Miękisz and C. Radin, The unstable chemical structure of the quasicrystalline alloys, Phys. Lett. 119A:133 (1986).
J. Miękisz, Quasicrystals—Microscopic Models of Nonperiodic Structures (Leuven, Lecture Notes in Mathematical and Theoretical Physics, Vol. 5, Leuven University Press, 1993).
W. Holsztynski and J. Slawny, Peierls condition and number of ground states, Commun. Math. Phys. 61:177 (1978).
J. Slawny, Low temperature properties of classical lattice systems: Phase transitions and phase diagrams, in Phase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987).
C. Radin and L. S. Schulmann, Periodicity and classical ground states, Phys. Rev. Lett. 51:621 (1983).
J. Miękisz and C. Radin, Third law of thermodynamics, Mod. Phys. Lett. 1B:61 (1987).
J. Miękisz, A global minimum of energy is not always a sum of local minima—a note on frustration, J. Stat. Phys. 71:425 (1993).
R. Penrose, Tilings and quasi-crystals; a non-local growth problem? in Introduction to the Mathematics of Quasicrystals Vol. 2 of Aperiodicity and Order, M. V. Jaric, ed. (Academic Press, 1989).
F. Gähler and H-C. Jeong, Quasicrystalline ground states without matching rules, J. Phys. A: Math. Gen. 28:1807 (1995).
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Miekisz, J. An Ultimate Frustration in Classical Lattice-Gas Models. Journal of Statistical Physics 90, 285–300 (1998). https://doi.org/10.1023/A:1023264004272
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DOI: https://doi.org/10.1023/A:1023264004272