Short-ranged resonating valence bond physics, quantum dimer models, and Ising gauge theories

Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising ${(Z}_2)$ gauge theories, but that in these limits the system develops a larger local U(1) invariance that has different consequences on different lattices. Conversely, we note that the standard $Z_2$ gauge theory is a generalized quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the U(1) down to $Z_2.$ These mappings indicate that at least one realization of the Senthil–Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable and a natural constraint. We also discuss the notion of topological order in $Z_2$ gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured $Z_2$ gauge field.