A programming guide for tensor networks with global [formula omitted] symmetry

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 adapting typical functions for symmetric tensors. 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. This fusion tree approach is readily extensible to anyonic systems, as we demonstrate for a chain of Fibonacci anyons.