Tensor-Network study of Ising model on infinite hyperbolic dodecahedral lattice

TL;DR

This paper introduces a tensor-network algorithm to study the Ising model on an infinite hyperbolic lattice, revealing a mean-field phase transition consistent with theoretical predictions.

Contribution

The authors extend the CTMRG algorithm from 2D to 3D and adapt it for hyperbolic lattices with dodecahedral cells, enabling analysis of infinite-dimensional structures.

Findings

Identified a continuous phase transition with mean-field critical exponents.

Estimated the phase transition temperature accurately.

Confirmed the universality class matches theoretical predictions.

Abstract

We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous non-critical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents $\beta=0.4999$ and $\delta=3.007$, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multi-state spin models.