of the set of the optimal free variational parameters $W^*_p(\theta_j)$ also provides helpful information about the phase transition $h_t^{(p)}$. The pairs of the optimal variational parameters $W^*_p(\theta_j)$ coupled by spin-inversion symmetry smoothly collapse onto a single curve exactly at the phase transition for all considered lattice geometries. This process follows the identical behavior as we have presented in [@DaniskaGendiar]. However, due to the large number of the variational parameters $N_{\rm Ising}^{(p)}$, we do not plot the $h$-dependence of $W^*_p(\theta_j)$ since the behavior remains qualitatively unchanged.

![The magnetic susceptibility $\chi_p$ of the TFIM as a function of the magnetic field $h$ for the hyperbolic lattices with $p \in \{5, \dots,10\}$. The vertical dot-dashed lines serve as guides for the eye and correspond to the phase transitions $h_t^{(p)}$.[]{data-label="Fig6"}](Fig6.pdf){width="70.00000%"}

Conclusions
===========

We have investigated three quantum spin-$\frac{1}{2}$ models (Heisenberg, XY, and TFIM) on a series of hyperbolic lattices by means of the numerical algorithm Tensor Product Variational Formulation [@DaniskaGendiar]. The series of lattices is constructed by tessellation of regular $p$-sided polygons with the fixed coordination number equal to four, where $p \in \{5, \dots,11\}$. The Euclidean square lattice ($p=4$) has been also considered as a reference lattice, although we have discussed in [@DaniskaGendiar] 