@lsh], which has been established as a universal marker to detect quantum phase transitions in many-body lattice systems. Indeed, the ground state fidelity per lattice site is closely related to the GE. Therefore, it is natural to expect that there should be an efficient way to compute the GE in the context of tensor network algorithms. This has been achieved for quantum many-body lattice systems with periodic boundary conditions in one spatial dimension in the context of the matrix product state representation [@hu].

Quantum phase transitions in 2D quantum lattice models can be investigated using a number of different physical quantities, including local order parameters, fidelity per lattice site, single-copy entropy and multi-partite entropy measurement, and GE per lattice site. As is well known, local order parameters are defined relating to the spontaneous symmetry breaking of some symmetry group, which results in degeneracy of ground states. Different degenerate ground states can be distinguished from different values of the local order parameters, and continuous and discontinuous phase transitions can be identified from the continuous and discontinuous behavior of the local order parameters. Although demonstrated to be capable of detecting phase transitions, it is not entirely clear if GE can distinguish different 