for unentangled states the GE is zero.

For our purpose, we shall consider a quantum many-body system on an infinite-size square lattice, which undergoes a quantum phase transition at a critical point in the thermodynamic limit. In this situation, each lattice site constitutes a party, thus the GE per party is the GE per lattice site, which is well defined in the thermodynamic limit ($N \rightarrow \infty$), since the contribution to the fidelity from each party (site) is multiplicative. In the infinite size limit we thus denote $${\mathcal E}=\lim_{N\to\infty} {\mathcal E}_{N}$$ as the GE per site.

geometric entanglement in infinite square lattice systems
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In this section we provide an analysis from the perspective of GE of different 2D quantum spin systems undergoing different types of quantum phase transitions. The GE per site has previously been applied to ground states of a number of 1D spin chains [@wei; @ow; @central; @scopy; @qqs; @jean; @hu; @hl] and some 2D models [@hl; @gehlw; @thesis; @roman; @z3pott] across primarily continuous phase transitions. Our analysis here extends these previous studies to a wider range of 2D quantum models and with more exotic situations. Previously, the GE of ground states for the quantum Ising model 