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Geometric properties of loop condensed phases on the square lattice

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.