Abstract
A problem in the theory of liquid crystals is to construct a model system which at low temperatures displays long-range orientational order, but not translational order in all directions. We present five lattice models (two two-dimensional and three three-dimensional) of hard-core particles with attractive interactions and prove (using reflection positivity and the Peierls argument) that they have orientational order at low temperatures; the two-dimensional models have no such ordering if the attractive interaction is not present. We cannot prove that these models do not have complete translational order, but their zero-temperature states are such that we are led to conjecture that complete translational order is always absent.
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Work of EHL supported by U.S. National Science Foundation Grant MCS 75-21684 A02. Financial assistance from the Danish Natural Science Research Council is also gratefully acknowledged.
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Heilmann, O.J., Lieb, E.H. Lattice models for liquid crystals. J Stat Phys 20, 679–693 (1979). https://doi.org/10.1007/BF01009518
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DOI: https://doi.org/10.1007/BF01009518