Ian Jauslin

Liquid crystals and the Heilmann-Lieb model

AMS Falls Eastern Sectional Meeting: Special session on Recent Probabilistic Advances in Mathematical Physics

October 10, 2020

A liquid crystal is a phase of matter in which order and disorder coexist: for some degrees of freedom, there is order, whereas for others, disorder. Such materials were discovered in the late XIXth century, but it took over a century to understand, from microscopic models, how such phases form. In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of such a liquid crystal phase. In this setting, this amounts to showing that dimers spontaneously align, but do not fully crystallize: there is no translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. In a joint work with E.H. Lieb, we have proved this conjecture, thus showing the spontaneous emergence of a liquid crystal phase in the Heilmann-Lieb model.

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