Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D

Abstract

A result by Gioia and Wang [Phys Rev X 12, 031007 (2022)] showed that translationally symmetric states having nonzero momentum are necessarily long range entangled (LRE). Here, we consider the question: can a notion of momentum for non-translation symmetric states directly encode the nature of their entanglement, as it does for translation symmetric states? We show the answer is affirmative for 1D systems, while higher dimensional extensions and topologically ordered systems require further work. While Gioia and Wang's result applies to states connected via finite depth quantum circuits to a translation symmetric state, it is often impractical to find such a circuit to determine the nature of the entanglement of states that break translation symmetry. Here, instead of translation eigenstates, we focus on the many-body momentum distribution and the expectation value of the translation operator in many-body states of systems having broken translation symmetry. We show that in the continuum limit the magnitude of the expectation value of the translation operator $|<T>|$ necessarily goes to $1$ for delocalized states, a proxy for LRE states in 1D systems. This result can be seen as a momentum-space version of Resta's formula for the localization length. We investigate how accurate our results are in different lattice models with and without well-defined continuum limits. To that end, we introduce two models: a deterministic version of the random dimer model, illustrating the role of the thermodynamic and continuum limits for our result at a lattice level, and a simplified version of the Aubry-Andre model, with commensurate hopping for both momentum and position space. Finally, we use the random dimer model as a test case for the accuracy of $|<T>|$ as a localization (and thus entanglement) probe for 1D periodic lattice models without a well-defined continuum limit.