Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (non-bipartite or
Z2) bosonic RVB states we studied, indicating that they are gapped
Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or Z2) bosonic RVB states we studied, indicating that they are gapped Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
The concept of the resonating valence bond (RVB) state was originally proposed by Anderson (1987) for the ground state of the positive-U Hubbard model. However, the on-site pairing long-range order is expected to exist in the ground state of the negative-U Hubbard model and to be incompatible with the RVB order. In this article, we shall rigorously prove that, in fact, these two long-range orderings either coexist or are suppressed simultaneously in the global ground states of the doped Hubbard models.
Using numerical results from a density matrix renormalization group study as a guide, we develop a resonating valence bond (RVB) theory for coupled Heisenberg chains. We argue that simple topological effects mandate a short-range RVB description of systems with an even number of chains
nc, with a spin gap, short-range correlations, and confinement of topological spin defects. Odd-
nc systems have long-range RVB ground states, no gap, and power-law correlations.
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). The latter system also corresponds to the ground state of the Rokhsar-Kivelson quantum dimer model at its critical point. 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 $1/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.20$, compared with $\ensuremath{\alpha}=2$ for the CDM). We also compute the distribution of monomer (defect) pair separations, which decay by a larger exponent in the RVB than in the CDM. We further study both models in their different winding-number sectors and evaluate the relative weights of different sectors. 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. Four independent measurements consistently show a value ${K}_{\mathrm{RVB}}\ensuremath{\approx}1.6{K}_{\mathrm{CDM}}$, with the same kinds of numerical evaluations of
KCDM giving results in agreement with the rigorously known value ${K}_{\mathrm{CDM}}=\ensuremath{\pi}/16$. 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. 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.