Parallel loop cluster quantum Monte Carlo simulation of quantum magnets based on global union-find graph algorithm

A large-scale parallel loop cluster quantum Monte Carlo simulation is presented. On 24,576 nodes of the K computer, one loop cluster Monte Carlo update of the world-line configuration of the $S=1/2$ antiferromagnetic Heisenberg chain with $2.6 \times 10^6$ spins at inverse temperature $3.1 \times 10^5$ is executed in about 8.62 seconds, in which global union-find cluster identification on a graph of about 1.1 trillion vertices and edges is performed. By combining the nonlocal global updates and the large-scale parallelization, we have virtually achieved about $10^{13}$-fold speed-up from the conventional local update Monte Carlo simulation performed on a single core. We have estimated successfully the antiferromagnetic correlation length and the magnitude of the first excitation gap of the $S=4$ antiferromagnetic Heisenberg chain for the first time as $\xi = 1.040(7) \times 10^4$ and $\Delta = 7.99(5) \times 10^{-4}$, respectively.