If the spins are compatible with the ↑, ↓ singlet restriction, this update can be accepted. (a) A simple two-bond update: Choosing two sites on the same sublattice, the two bonds connected to them can be reconfigured in a unique way. In the basic moves (a) and (b), only one of the two valence-bond configurations is affected at a time. Open and solid circles represent ↑ and ↓ spins. In addition to the standard case of short bonds only, we also studied ensembles in which fourth-neighbor (bipartite) bonds are allowed at a density controlled by a tunable fugacity, resulting (as expected) in a smooth reduction of K.įigure 4Monte Carlo updates for the RVB state in the combined spin-bond basis.
The background of a nonzero winding-number gradient W / L introduces spatial anisotropies and an increase in the effective K, both of which can be understood as a consequence of anharmonic terms in the height-model free energy, which are of relevance to the recently proposed scenario of “Cantor deconfinement” in extended quantum dimer models. Four independent measurements consistently show a value K RVB ≈ 1.6 K CDM, with the same kinds of numerical evaluations of K CDM giving results in agreement with the rigorously known value K CDM = π / 16. Like the CDM, all the observed RVB behaviors can be understood in the framework of a mapping to a “height” model characterized by a gradient-squared stiffness constant K. We further study both models in their different winding-number sectors and evaluate the relative weights of different sectors. We also compute the distribution of monomer (defect) pair separations, which decay by a larger exponent in the RVB than in the CDM. We find that, although spin-spin correlations decay exponentially in the RVB, four-spin valence-bond-solid correlations are critical, qualitatively like the well-known dimer-dimer correlations of the CDM, but decaying more slowly (as 1 / r α with α ≈ 1.20, compared with α = 2 for the CDM).
The latter system also corresponds to the ground state of the Rokhsar-Kivelson quantum dimer model at its critical point. We use Monte Carlo simulations to study properties of Anderson's resonating-valence-bond (RVB) spin-liquid state on the square lattice (i.e., the equal superposition of all pairing of spins into nearest-neighbor singlet pairs) and compare with the classical dimer model (CDM).
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