  The R\'enyi entanglement entropy in quantum many-body systems can be viewed
as the difference in free energy between partition functions with different
trace topologies. We introduce an external field $\lambda$ that controls the
partition function topology, allowing us to define a notion of nonequilibrium
work as $\lambda$ is varied smoothly. Nonequilibrium fluctuation theorems of
the work provide us with statistically exact estimates of the R\'enyi
entanglement entropy. This framework also naturally leads to the idea of using
quench functions with spatially smooth profiles, providing us a way to average
over lattice scale features of the entanglement entropy while preserving long
distance universal information. We use these ideas to extract universal
information from quantum Monte Carlo simulations of SU(N) spin models in one
and two dimensions. The vast gain in efficiency of this method allows us to
access unprecedented system sizes up to 192 x 96 spins for the square lattice
Heisenberg antiferromagnet.
