Solving the quantum dimer and six vertex models one electric field line at a time

The nature and the very existence of the resonant plaquette valence bond state that separates the classical columnar phase and the Rokhsar and Kivelson point in the quantum dimer model remains unsettled. Here we take a different line of attack on this model, and on the closely related six vertex model, by exploiting the global conservation law of the number of electric field lines. This allows us to study a single fluctuating electric field line which we show maps exactly onto a one dimensional spin chain. In the case of the six vertex model, the electric field line maps onto the celebrated spin 1/2 XXZ model which can be solved exactly. In the quantum dimer model, the electric field line is mapped onto a two-leg spin 1/2 ladder, which we study using numerical exact diagonalization. Our findings are consistent with the existence of three distinct phases including a Luttinger liquid phase, the one-dimensional precursor to the two-dimensional plaquette valence bond solid. The uncanny resemblance of our quasi-one-dimensional electric field line problem to the full two-dimensional problem suggests that much of the behavior of the latter might be understood by thinking of it as a closely packed array of field lines which themselves are undergoing non-trivial phase transitions.