in a transverse field and the XYX model on the square lattice have been investigated with TRG [@hl] and iPEPS [@gehlw] approaches, both of which show a continuous behavior in the GE corresponding to a continuous phase transition. Our results are based on the application and improvement of an iPEPS algorithm for the calculation of the GE of 2D quantum systems [@gehlw], as detailed in Appendices A and B. Preliminary results along this direction were also reported in Ref. [@thesis]. The size of the truncation dimension $D$ controls the underlying accuracy of the iPEPS algorithm. For the purposes of this study, which focusses on the general behavior of the GE in the vicinity of quantum critical points, we consider sufficiently large values of $D$, compared to Refs. [@gehlw; @thesis], to be confident of the observed behavior. Refinements in the implementation of the basic iPEPS algorithm have been discussed recently [@tn_latest].

2D quantum Ising model in a transverse field
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We consider the hamiltonian $$H = -\sum_{( \vec r, \vec r')}
     S^{[\vec r]}_x S^{[\vec r']}_x - \lambda \sum_{\vec r}  S^{[\vec r]}_z,$$ where here and in later subsections $S^{[\vec r]}_{\alpha}(\alpha=x,y,z)$ are spin-$\frac12$ Pauli operators acting at site $\vec r$, with $( \vec 