  Loop condensed phases are scale-invariant quantum liquid phases of matter.
These phases include topologically ordered liquid phases such as the toric code
as well as critical liquids such as the Rokhsar-Kivelson point of the quantum
dimer model on the square lattice. To investigate the extent to which nonlocal
geometric observables capture a signature of the nonlocal quantum order present
in these phases, we compute geometric properties of such loop condensed states
using directed loop Monte Carlo calculations. In particular, we investigate the
loop condensed nature of ground state of the square lattice quantum dimer model
at the Rokhsar-Kivelson point and compare with other loop condensed states on
the square lattice, including those of the toric code and fully packed loop
model. The common features of such liquids are a scale invariant distribution
of loops and a fractal dimensionality of spanning loops. We find that the
fractal dimension of the loop condensate of the square lattice quantum dimer
model at the Rokhsar-Kivelson point is 3/2, which provides quantitative
confirmation of the effective height model that is commonly used to describe
this critical dimer liquid.
