Cluster Algorithms for General-SQuantum Spin Systems

We present a general strategy to extend quantum cluster algorithms for $S=\frac{1}{2}$ spin systems, such as the loop algorithm, to those with an arbitrary size of spins. The partition function of a high- $S$ spin system is generally represented by the path integral of a $S=\frac{1}{2}$ model with special boundary conditions in the imaginary-time direction. We introduce additional graphs for the boundary part and give the labeling probability explicitly, which completes the algorithm together with an existing $S=\frac{1}{2}$ algorithm. As a demonstration, we simulate the integer-spin antiferromagnetic Heisenberg chains. The magnitude of the first excitation gap is estimated to be 0.41048(6), 0.08917(4), and 0.01002(3) for $S=1\mathrm{}$, 2, and 3, respectively.