  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.
