Critical Correlations for Short-Range Valence-Bond Wave Functions on the Square Lattice

We investigate the arguably simplest $SU(2)$-invariant wave functions capable of accounting for spin-liquid behavior, expressed in terms of nearest-neighbor valence-bond states on the square lattice and characterized by different topological invariants. While such wave-functions are known to exhibit short-range spin correlations, we perform Monte Carlo simulations and show that four-point correlations decay algebraically with an exponent $1.16(4)$. This is reminiscent of the {\it classical} dimer problem, albeit with a slower decay. Furthermore, these correlators are found to be spatially modulated according to a wave-vector related to the topological invariants. We conclude that a recently proposed spin Hamiltonian that stabilizes the here considered wave-function(s) as its (degenerate) ground-state(s) should exhibit gapped spin and gapless non-magnetic excitations.