Other values for \(N_\sigma\) and \(\xi\)

In this appendix, we report the result of additional simulations for different values of \(N_\sigma\) and \(\xi\). For the \(C_{2v}\) model, we show how at higher couplings, \(\xi\), discontinuities occur due to level crossings, as already described in Section 2.3. For the \(C_{4v}\) model, we show that the transition stays continuous for all considered parameters.

The \(C_{2v}\) model

Fig. 3 shows the structure factor correlation ratio and derivative of free energy for the \(C_{2v}\) model at \(N_\sigma=4\) and \(\xi \in \{0.25, 0.4, 0.5\}\). For these parameters we observe a continuous phase transition. Fig. 4 plots the same observables for \(N_\sigma=2\). For lower values of the coupling \(\xi\) the curves are also smooth, but at \(\xi =0.5\) discontinuities appear, which get more pronounced at \(\xi=0.75\). At \(\xi=0.75\), one can observe multiple discontinuities for a single system size, e.g. at \(h\approx 4.2\) and \(h\approx 4.4\) for \(L=20\). These discontinuities occur due to level crossings, as already described in the mean field part in Section 2.3. As shown in Fig. 4(d,f) and elaborated in the mean field section, they can be avoided by twisting the boundary conditions of the fermionic degrees of freedom.

Fig. 3 Structure factor correlation ratio and derivative of free energy for the \(C_{2v}\) model at \(N_\sigma=4\) and \(\xi \in \{0.25, 0.4, 0.5\}\). The data is consistent with continuous transitions.

Fig. 4 Structure factor correlation ratio and derivative of free energy for the \(C_{2v}\) model at \(N_\sigma=2\) and \(\xi \in \{0.25, 0.4, 0.5, 0.75\}\). At \(\xi=0.5\) and \(\xi=0.75\) discontinuities due to level crossing emerge. They can be avoided by twisting the boundary conditions of the fermionic degrees of freedom.

The \(C_{4v}\) model

Fig. 5 and Fig. 6 have the same layout as the previous figures and show only continuous transitions for various combinations of \(N_\sigma \in \{1, 2\}\), \(\xi \in \{0.5, 0.75, 1, 2\}\). We also show data at \(\xi = 0\), which corresponds to the transverse-field Ising model.

Fig. 5 Structure factor correlation ratio and derivative of free energy for the \(C_{4v}\) model at \(N_\sigma=1\) and \(\xi \in \{0, 1, 2\}\). The data shows continuous transitions.

Fig. 6 Structure factor correlation ratio and derivative of free energy of the \(C_{4v}\) model at \(N_\sigma=2\) and \(\xi \in \{0.5, 0.75, 2\}\). The data shows continuous transitions.