A programming guide for tensor networks with global $SU(2)$ symmetry

TL;DR

This paper provides practical programming tips and data structures for implementing tensor network algorithms with global $SU(2)$ symmetry, emphasizing scalable methods for complex, high-dimensional systems.

Contribution

It introduces a practical approach to implement $SU(2)$ symmetry in tensor networks using fusion trees, avoiding explicit storage of structural tensors.

Findings

Fusion trees efficiently encode $SU(2)$ symmetry constraints.

Implementation scales well for high-dimensional tensor network simulations.

Provides detailed guidance on data structures and tensor manipulations.

Abstract

This paper is a manual with tips and tricks for programming tensor network algorithms with global $SU(2)$ symmetry. We focus on practical details that are many times overlooked when it comes to implementing the basic building blocks of codes, such as useful data structures to store the tensors, practical ways of manipulating them, and so forth. Here we do not restrict ourselves to any specific tensor network method, but keep always in mind that the implementation should scale well for simulations of higher-dimensional systems using, e.g., Projected Entangled Pair States, where tensors with many indices may show up. To this end, the structural tensors (or intertwiners) that arise in the usual decomposition of $SU(2)$-symmetric tensors are never explicitly stored throughout the simulation. Instead, we store and manipulate the corresponding fusion trees - an algebraic specification of the symmetry constraints on the tensor - in order to implement basic $SU(2)$-symmetric tensor operations.