Tensor network states and algorithms in the presence of a global U(1) symmetry

TL;DR

This paper discusses how to incorporate U(1) symmetry into tensor network algorithms, enabling exact particle number conservation and reducing computational costs in many-body quantum simulations.

Contribution

It specializes previous work to U(1) symmetry, detailing tensor decomposition and manipulation techniques that improve efficiency and accuracy in tensor network methods.

Findings

U(1) symmetric tensors enable exact particle number conservation.

Use of symmetry reduces computational costs.

Application demonstrated in multi-scale entanglement renormalization ansatz.

Abstract

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.