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 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 nonmagnetic excitations.