Thermal Order by Disorder in RVB States on the Checkerboard Lattice I: Analytical Considerations

RVB states have shaped our understanding of spin liquids. Short-range RVBs are closely related to quantum dimer models, yet the nature of spin correlations can be quite different in these states due to the non-orthogonality of different RVB states. A classic example is provided by nearest-neighbor RVBs on the square lattice. Whereas correlations between quantum dimers are characterized by the power-law decay, the spin-spin correlations in the corresponding RVB state fall off exponentially with distance. However, the singlet-singlet correlations still obey a power law which is different from quantum dimers. Conversly, on non-bipartite lattices such as triangular, both dimer and RVB liquids exhibit exponentially decaying correlations.

We focus on the RVBs on the checkerboard lattice with one spin singlet in each plaquette and show that they do not fit this pattern. Whereas correlations between quantum dimers would obey a power law of the quantum 6-vertex model, the superposition of spin singlets are different. Different valence bond states in the superposition cannot have the same sign; their interference results in the exponential decay of both spin-spin and singlet-singlet correlations. When off-diagonal density-matrix elements are suppressed, e.g. by thermal decoherence, the singlet–singlet correlations become power-law. This model provides an unusual example of decoherence leading to qualitatively longer correlations between spin singlets in a spin-liquid like state.