Topological order in two-dimensional (2D) quantum matter can be determined by
the topological contribution to the entanglement Rényi entropies. However,
when close to a quantum phase transition, its calculation becomes cumbersome.
Here, we show how topological phase transitions in 2D systems can be much
better assessed by multipartite entanglement, as measured by the topological
geometric entanglement of blocks. Specifically, we present an efficient tensor
network algorithm based on projected entangled pair states to compute this
quantity for a torus partitioned into cylinders and then use this method to
find sharp evidence of topological phase transitions in 2D systems with a
string-tension perturbation. When compared to tensor network methods for Rényi
entropies, our approach produces almost perfect accuracies close to
criticality and, additionally, is orders of magnitude faster. The method can
be adapted to deal with any topological state of the system, including
minimally entangled ground states. It also allows us to extract the critical
exponent of the correlation length and shows that there is no continuous
entanglement loss along renormalization group flows in topological phases
