Generalization of the tensor renormalization group approach to 3-D or higher dimensions

TL;DR

This paper extends the tensor renormalization group method to three or higher dimensions by establishing a mathematical connection with polytope geometry and introducing tensor decomposition techniques, verified on the 3-D Ising model.

Contribution

It introduces a novel generalization of TRG to higher dimensions, linking it with polytope geometry and tensor decomposition, supported by initial numerical tests.

Findings

Theoretical framework for 3-D tensor contraction

Connection between TRG patterns and polytope truncation

Numerical verification on 3-D Ising model

Abstract

In this paper, a way of generalizing the tensor renormalization group(TRG) is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is discovered. A theoretical contraction framework is therefore proposed. Furthermore, the canonical polyadic decomposition is introduced to tensor network theory. A lowest order numerical verification of this method on the 3-D Ising model is carried out.