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

Abstract: 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 enough 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 refined method of targeting single-particle excitations inside 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.