the mean-field universality class, because the hyperbolic lattices exhibit the infinite Hausdorff dimension, which significantly exceeds the critical lattice dimension $D_c=3$ [@Baxter]. Because of the mean-field-like character of the TPS approximation, the TPVF algorithm is expected to be more accurate whenever a hyperbolic lattice geometry is considered [@DaniskaGendiar; @hctmrg-Ising-p-4].

![The ground-state energy $E_0^{(p)}$ of the XY model with respect to the lattice parameter $p \in \{4,5,\dots, 11\}$. The inset shows the zoomed-in energy including the details of the fitting function.[]{data-label="Fig2"}](Fig2.pdf){width="70.00000%"}

![The ground-state energy $E_0^{(p)}$ of the Heisenberg model with respect to the lattice parameter $p \in \{4,5,\dots, 11\}$. The fitting function parameters are shown in the inset.[]{data-label="Fig3"}](Fig3.pdf){width="70.00000%"}

Figure \[Fig2\] illustrates a monotonically increasing and rapidly saturating curve of the energy $E_0^{(p)}$ for the XY model with respect to the lattice parameter $p$. The inset depicts the tail of the curve in detail together with an exponential fit applied to the five energies $E_0^{(7)}, \dots, E_0^{(11)}$. The case $p=4$, where the TPVF algorithm is not sufficiently accurate for the reasons mentioned above, was excluded from the extrapolation analysis. The fitting function is proposed in the form $$E_0^{(p)} = E_0^{(\infty)} +a_1 \exp(a_2 p ) \, ,
 \label{fitXY}$$ where $E_0^{(\infty)}$, $a_1$, and $a_2$ 