of the numerical results is presented in Sec. III and we summarize them in Sec. IV.

The Model
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We study the ground-state properties and the phase transition of the quantum TFIM, XY, and Heisenberg models in the thermodynamic limit on a series of hyperbolic lattices. Each hyperbolic lattice is made from equivalent congruent $p$-sided polygons. The polygon vertices coincide with the lattice spin sites, where a single spin is positioned. Each spin site has four nearest-neighboring spin sites, which is commonly referred to as the coordination number equal to four. We investigate the three models on a set of regular hyperbolic lattices of infinite size with the lattice parameter $p \in \{5,6,\dots, 11\}$. Apart from the set, we include two additional cases: $p=4$ being the Euclidean square lattice and the asymptotic case $p\to\infty$, which is associated to the Bethe lattice. Figure \[Fig1\] depicts the typical structure of the lattices. The square lattice serves as a reference lattice. The three spin models and the TPVF algorithm have been described in detail in [@DaniskaGendiar], and we focus only on the substantial aspects of the models on the hyperbolic lattices in the following.

In general, the Hamiltonian ${\cal H}$ of 