  We develop a nonequilibrium increment method in quantum Monte Carlo
simulations to obtain the R\'enyi entanglement entropy of various quantum
many-body systems with high efficiency and precision. To demonstrate its power,
we show the results on a few important yet difficult $(2+1)d$ quantum lattice
models, ranging from the Heisenberg quantum antiferromagnet with spontaneous
symmetry breaking, the quantum critical point with O(3) conformal field theory
(CFT) to the toric code $\Z_2$ topological ordered state and the Kagome $\Z_2$
quantum spin liquid model with frustration and multi-spin interactions. In all
these cases, our method either reveals the precise CFT data from the
logarithmic correction or extracts the quantum dimension in topological order,
from the dominant area law in finite-size scaling, with very large system
sizes, controlled errorbars and minimal computational costs. Our method
therefore establishes a controlled and practical computation paradigm to obtain
the difficult yet important universal properties in highly entangled quantum
matter.
