Sign-problem-free effective models of triangular lattice quantum antiferromagnets

The triangular lattice antiferromagnet with $S=1/2$ spins and nearest neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of 120 degrees. Numerical studies of quantum phases proximate to this state have been limited to small systems because the of the sign-problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order, and a sign-problem free Monte Carlo algorithm, enabling studies in large systems sizes. The model is a $\mathbb{Z}_2$ gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e. there is a static, background $\mathbb{Z}_2$ gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36 \times 36 \times 36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12} \times \sqrt{12}$ unit cell, and a gapped $\mathbb{Z}_2$ spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.