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 $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because 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 $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ 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.