Elementary excitations in gapped quantum spin systems

TL;DR

This paper demonstrates that in gapped quantum spin systems, localized operators can approximate excited states with specific momentum and energy, with accuracy improving as the operator's support grows, leveraging Lieb-Robinson bounds.

Contribution

It establishes a method to approximate excited eigenstates in gapped quantum systems using local operators, extending understanding of elementary excitations.

Findings

Approximation error decreases with support size of local operators.

Explicit demonstration on the AKLT model.

Discussion of generalizations and potential applications.

Abstract

For quantum lattice systems with local interactions, the Lieb-Robinson bound acts as an alternative for the strict causality of relativistic systems and allows to prove many interesting results, in particular when the energy spectrum exhibits an energy gap. In this Letter, we show that for translation invariant systems, simultaneous eigenstates of energy and momentum with an eigenvalue that is separated from the rest of the spectrum in that momentum sector, can be arbitrarily well approximated by building a momentum superposition of a local operator acting on the ground state. The error decreases in the size of the support of the local operator, with a rate that is set by the gap below and above the targeted eigenvalue. We show this explicitly for the AKLT model and discuss generalizations and applications of our result.