Institut de Physique Théorique
Unité de Recherche Associée au CNRS
Classical dimer models
In two dimensionsThe study of quantum dimer models (QDM) have greatly benefited from the good understanding of some of their classical analogs (works of Kasteleyn and Fisher in the 60's). However, very little was known about classical dimer models in presence short-ranged interaction beyond the hard-core repulsion, although such interactions are crucial in the QDM studied so far. In the refs. [1] and [3] below, we studied numerically (Monte-Carlo and transfer-matrix calculations) the simplest classical dimer model on the square lattice with nearest-neighbor dimer interaction (favoring dimer alignment). This model turned out to have two phases: a critical phase at high temperature described by a dilute Coulomb gas (``rough" in the height model terminology) and a dimer crystal at low temperature (``flat" in a height representation). The transition between the two phases is Kosterliz-Thouless like. We provided accurate estimates for the Coulomb gas coupling constant and critical exponents as a function of temperature (critical phase). The introduction of monomers (doping) and the implications of the finite-temperature phase diagram of the square-lattice QDM are also discussed.
In three dimensionsA first step in the study of three-dimensional dimer models (3D analog of the model above, on the cubic lattice, see Fig. on the right) was carried out by Huse et al. (2003) who identified, at infinite temperature, a "Coulomb" liquid, with no broken symmetry and algebraic (1/r3) dimer-dimer correlations. We extended their study by investigating the phase diagram of the model as a function of temperature by means of intensive Monte-Calro simulations (refs. [2,4] below). Due to the parallel-dimer-attraction (shaded plaquette in the figure), the system orders at low temperature and forms a dimer crystal. Although Landau-type arguments would generically predict a first order transition between these to phases, our numerical results strongly indicate that it is a continous one, with critical exponents compatible with that of a tricritical point. Similarities and differences with the "non-Landau-Ginzburg-Wilson" phase transitions, originally proposed for some 2D quantum or 3D classical spin systems [Senthil et al. (2004), Motrunich and Vishwanath (2004)] are discussed.
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