r, \vec r')$ running over all possible nearest-neighbor pairs on the square lattice. The parameter $\lambda$ measures the strength of the transverse magnetic field, with a phase transition from a ferromagnetic phase with two degenerate ground states to a paramagnetic phase with a single ground state occurring at the critical value $\lambda_c$. This model has been widely studied via a number of different techniques. Of particular relevance here are the previous calculation of the GE using the tensor product state and tensor renormalization group approach [@hl] and iPEPS [@gehlw; @thesis].

![(color online) Numerical values for (a) the local order parameter $\langle S_x\rangle$ and (b) the GE per site ${\mathcal E}_{\infty}(\lambda)$ for the 2D quantum Ising model as a function of the transverse field strength $\lambda$ for the indicated values of the iPEPS truncation dimension $D$. []{data-label="fig1"}](q2a.eps "fig:"){width="0.6\linewidth"} ![(color online) Numerical values for (a) the local order parameter $\langle S_x\rangle$ and (b) the GE per site ${\mathcal E}_{\infty}(\lambda)$ for the 2D quantum Ising model as a function of the transverse field strength $\lambda$ for the indicated values of the iPEPS truncation dimension $D$. []{data-label="fig1"}](q2b.eps "fig:"){width="0.7\linewidth"}

For different values of the truncation dimension $D$, when $\lambda<\lambda_c(D)$, spontaneous symmetry breaking occurs with $S_x$ 