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Generic mixed columnar-plaquette phases in Rokhsar-Kivelson models

Arnaud Ralko, Didier Poilblanc and Roderich Moessner

The Rokhsar-Kivelson Quantum Dimer model (RK-QDM) on the square lattice, originally proposed in the context of high-temperature superconductivity, and its descendants have taken on a central role in the study of quantum systems incorporating a hard local constraint. They have thus been prominent in the context of hardcore bosons hopping on frustrated lattices, Josephson junction arrays, frustrated Ising models in a transverse field or with small XY exchange, gauge theories in unusual sectors, spin orbital models and cold atoms. Studies of this model and its extensions have unearthed a wealth of phenomena, including instances of deconfined quantum criticality and a new route to deconfinement.

As many RK models belong to the rare class of models of correlated quantum matter without a sign problem, they are in principle amenable to efficient numerical study ; they also tend to have well-studied effective field theories formulated in terms of height/gauge degrees of freedom.

In this work, we propose a generic alternative scenario to a first-order plaquette-columnar transition, as a continuous interpolation between them via a mixed phase (see Fig.1).

This we do by deriving from the dimer model a height model, with the height variable h, living on the plaquettes of the square lattice. Mutatis mutandis, such a mapping applies to d=2 RK models incorporating constraints which can be cast as local U(1) conservation laws. In this model, an ideal columnar phase has a mean value < h > equal to a half-integer, whereas an ideal plaquette phase has an integer < h >.

In this representation, the leading terms in an effective action incorporating all terms not ruled out by symmetry or microscopic considerations of the model in d=2+1 dimension are :

 S = \int d^{2} x d \tau [(\partial_\tau h)^2+ \rho_2 (\nabla h)^2 + \lambda \cos(2 \pi h)+\mu \cos(4 \pi h)]

where \tau denotes Euclidean time.

We find, depending on the value of the new term \mu, a low-symmetry mixed state which continuously crosses over from primarily columnar to plaquette character (see Fig.1, central picture).

We also give a confirmation of the presence of this anisotropic scenario involving the mixed phase in the QDM on the square lattice, by the use of the Green’s function Monte-Carlo algorithm and symmetry considerations, applied for calculating the factor structure and the first gap excitation spectra for representative order parameters.

Post-scriptum :

Details and references are given in the corresponding paper "Generic mixed columnar-plaquette phases in Rokhsar-Kivelson models" published in :

A. Ralko, D. Poilblanc and R. Moessner, Physical Review Letters 100, 037201 (2008).