Analysis of quantum spin models on hyperbolic lattices and Bethe lattice

TL;DR

This paper investigates quantum spin models on hyperbolic lattices using tensor network methods, revealing mean-field-like behavior and estimating properties on the Bethe lattice relevant to AdS/CFT correspondence.

Contribution

It introduces a tensor product variational formulation to analyze quantum spin models on hyperbolic lattices and estimates their properties on the Bethe lattice.

Findings

Hyperbolic lattice geometry induces mean-field-like universality.

Ground-state energies and phase transition fields are estimated for various models.

Results are relevant for understanding models in non-Euclidean geometries and AdS/CFT.

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 the mean-field-like universality 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.