Phase diagram of quantum square ice

We have investigated the ground-state and finite-temperature phase diagram of quantum square ice - realized by the transverse-field Ising model on a checkerboard lattice - using both linear spin-wave (LSW) theory and quantum Monte Carlo (QMC). We generalize the model with different couplings between nearest ($J_1$) and next-to-nearest ($J_2$) neighbors on the checkerboard lattice. Our QMC approach generalizes the loop algorithm - very efficient in the study of constrained classical systems - to a ``brane algorithm'' for quantum systems. At the LSW level the vast degeneracy of the ground-state for $J_1=J_2$ and $J_2>J_1$ remains intact; moreover LSW theory breaks down in extended regions of the phase diagram, pointing at non-classical states [1]. Our QMC study goes beyond perturbative schemes and addresses directly the nature of the low-temperature phases. We have critically examined the possibility of a resonating-plaquette state for $J_1=J_2$, suggested by degenerate perturbation theory on the ice-rule manifold for weak fields. Our QMC results for finite fields confirm the absence of N\'eel or collinear order, but they do not confirm the presence of resonating-plaquette order, pointing at a possibly more complex non-classical state.\\[4pt] [1] L.-P. Henry et al., PRB 85, 134427 (2012).