Automated Construction of U(1)-invariant Matrix-Product Operators from Graph Representations

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary U(1)-invariant operators in case a finite-states-machine (FSM) formulation exists. The method automatizes two major construction steps:
i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers;
ii) the appearance of phase factors due to commutation rules.
This is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of U(1)-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. We apply this ansatz to study the generation of arbitrary products of operators in terms of FSM, resulting in an exact MPO representations for the variance of the Hamiltonian of a S=1 Heisenberg chain.