---
abstract: 'The quantum XY, Heisenberg, and transverse field Ising models on hyperbolic lattices are studied by means of the Tensor Product Variational Formulation algorithm. The lattices are constructed by tessellation of congruent polygons with coordination number equal to four. The calculated ground-state energies of the XY and Heisenberg models and the phase transition magnetic field of the Ising model on the series of lattices are used to estimate the corresponding quantities of the respective models on the Bethe lattice. The hyperbolic lattice geometry induces mean-field-like behavior of the models. The ambition to obtain results on the non-Euclidean lattice geometries has been motivated by theoretical studies of the anti-de Sitter/conformal field theory correspondence.'
address: 'Institute of Physics, Slovak Academy of Sciences, SK-845 11, Bratislava, Slovakia'
author:
- Michal Daniška and Andrej Gendiar
title: Analysis of quantum spin models on hyperbolic lattices and Bethe lattice
---

[*Keywords*]{}: tensor product state, quantum spin systems, non-Euclidean geometry, phase transition

Introduction
============

Many analytical and computational techniques have been developed to study quantum spin models on two-dimensional (Euclidean) lattices. Among such techniques, let us mention the corner transfer matrix approach [@Baxter], the coordinate Bethe Ansatz [@Bethe], the algebraic Bethe Ansatz [@Fad], the vertex operator 