Quantum dimer models and effective Hamiltonians on the pyrochlore lattice

Abstract

We study a large-$N$ deformation of the $S=1∕2$ pyrochlore Heisenberg antiferromagnet which leads to a soluble quantum dimer model at leading nontrivial order. In this limit, the ground state manifold---while extensively degenerate---breaks the inversion symmetry of the lattice, which implies a finite temperature Ising transition without translational symmetry breaking. At lower temperatures and further in the $1∕N$ expansion, we discuss an effective Hamiltonian within the degenerate manifold, which has a transparent physical interpretation as representing dimer potential energies. We find mean-field ground states of the effective Hamiltonian which exhibit translational symmetry breaking. The entire scenario offers a new perspective on previous treatments of the SU(2) problem not controlled by a small parameter, in particular showing that a mean-field state considered previously encodes the physics of a maximally flippable dimer configuration. We also comment on the difficulties of extending our results to the SU(2) case, and note implications for classical dimer models.