Efficient and systematic calculation of arbitrary observables for the matrix product state excitation ansatz

Numerical methods based on matrix product states (MPSs) are currently the de facto standard for calculating the ground state properties of (quasi-) one-dimensional quantum many-body systems. While the properties of the low-lying excitations in such systems are often studied in this MPS framework through dynamics_ by means of time-evolution simulations, we can also look at their ₛtatics_ by directly calculating eigenstates corresponding to these excitations. The so-called MPS excitation ansatz is a powerful method for finding such eigenstates with a single-particle character in the thermodynamic limit. Although this excitation ansatz has been used quite extensively, a general method for calculating expectation values for these states is lacking in the literature: we aim to fill this gap by presenting a recursive algorithm to calculate arbitrary observables expressed as matrix product operators. This method concisely encapsulates existing methods for -- as well as extensions to -- the excitation ansatz, such as excitations with a larger spatial support and multi-particle excitations, and is robust to handle further innovations. We demonstrate the versatility of our method by studying the low-lying excitations in the spin-1 Heisenberg chain and the one-dimensional Hubbard model, looking at how the excitations converge in the former, while in the latter, we present a novel method of targeting single-particle excitations inside of a continuum by minimizing the energy ᵥariance_ rather than the energy itself. We hope that this technique will foster further advancements with the excitation ansatz.