$44$    $63$
   \[0.04cm\] $N_{\rm Ising}^{(p)}$    $6$   $8$   $13$   $18$   $30$   $46$   $78$   $126$
              \[0.04cm\]                                                            

  : The numbers of the free variational parameters $N_{\rm Heis}^{(p)}$ (for the XY and the Heisenberg models) and $N_{\rm Ising}^{(p)}$ (for the TFIM) including the normalization parameter.

\[Tab1\]

The free variational parameters $W_p(\theta_j)$ are optimized numerically by TPVF [@DaniskaGendiar]. It is based on the fact that the product structure of the state $|\Psi_p\rangle$ allows to calculate the numerator ${\cal N}(W_p(\{\sigma\}))$ and the denominator ${\cal D}(W_p(\{\sigma\}))$ in  for the given set of the tensor elements $W_p(\{\sigma\})$ by an appropriate modification of the CTMRG algorithm. Having applied the CTMRG as the effective and accurate numerical tool for calculation of the ratio in , a multi-dimensional minimizer is used for optimizing the variational parameters $W_p(\theta_j)$ [@gsl1; @gsl2; @NM].

Numerical results
=================

XY and Heisenberg models
------------------------

We study the XY and the Heisenberg models at zero magnetic field, where these models are known to be critical in the Euclidean space. Therefore, there is no preferred direction (the spin alignment) in the system on the Euclidean lattice at $h \geq 0$, and the spin-inversion symmetry is present. We expect that the models on hyperbolic lattices also exhibit the spin-inversion symmetry. 