Detecting topological order from modular transformations of ground states on the torus

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. Here we show that there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. These models share the same mutual statistics, and their self statistics differ by phase. We provide the upper limit of the information one may obtain from the overlap of MESs. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from the overlap of MESs.