if applied separately onto two sets: those with even $p \in \{6, 8, 10\}$ (the lower branch shown in the inset) and the odd $p \in \{5, 7, 9, 11\}$ (the upper branch). The fitting parameters of the two regressions are listed in the inset of figure \[Fig3\]. The lower and the upper branches yield the energies $E_0^{(\infty)}$ $-1.2919443$ and $E_0^{(\infty)}-1.2919440$, respectively. With respect to an independent application of additional analogous fits, we found $E_0^{(\infty)}=-1.291944$ (all the digits are valid) to be considered as the correct estimate of the ground-state energy per bond of the Heisenberg model on the Bethe lattice.

We have not found any theoretical reasoning for the exponential convergence of the ground-state energies $E_0^{(p)}$ yet. However, if a power-law fitting function was applied instead, we obtained a less accurate fitting and greater $RSS$.

Transverse field Ising model
----------------------------

The TFIM undergoes a quantum phase transition at a nonzero magnetic field $h_t^{(p)}>0$, where we explicitly emphasize its dependence on the lattice geometry. The nonzero spontaneous magnetization in the ordered phase at $h<h_t^{(p)}$ breaks the spin-inversion symmetry, which results in approximately twice larger set of the free variational parameters $N_{\rm Ising}^{(p)}$ in the TPVF algorithm if compared to $N_{\rm 