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

TL;DR

This work establishes a momentum-space analogue to Resta’s localization framework for one-dimensional systems that break translation symmetry, showing that the magnitude of the translation operator expectation $z_t=|\langle T\rangle|$ tracks the spread of the many-body momentum distribution and signals long-range entanglement in the continuum limit. By linking $C_2^{(K)}$ to $z_t$ through $C_2^{(K)}=-\tfrac{2}{a^2}\log z_t$ and employing the modular-variable uncertainty with $U=e^{i\frac{2\pi}{L}\hat X}$ and $T=e^{ia\hat K}$, the authors connect momentum-space delocalization to entanglement while clarifying the distinct roles of continuum and thermodynamic limits. They test the framework across several 1D lattice models, including the Deterministic Dimer Model, Simple Self-Dual model, Random Dimer Model, and Aubry–Andre model, and they analyze flux insertion to distinguish localized from delocalized states via phase sensitivity of $\langle T\rangle$. The study finds that $z_t$ robustly signals delocalization and LRE in translation-broken 1D systems, while $z_x$ remains a more natural probe for localization, with flux-phase information providing complementary insight; extensions to higher dimensions and topologically ordered phases remain open directions. Overall, the paper offers a practical, translation-property–agnostic method to infer entanglement structure from momentum-space observables in 1D systems.

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.

Figures (20)

• Figure 1: Illustration of sample momentum distributions for 1D systems having twisted periodic boundary conditions with twist $\phi$. Translation-symmetric states (first row) have delta-function-like momentum distributions. For SRE/localized states the peak is always centered at trivial momentum $K_0=0$ (or $\pi$, for fermions) even after twisting boundary conditions. For LRE/delocalized states, even a state that starts at a trivial value $K_0(\phi=0)=0$ will have its momentum shifted to $K_0(\phi)=\phi$ when subject to twisted boundary conditions. We can compare the distributions of the top row of translation symmetric states to the distributions for the translation-broken states in the bottom row. Localized states that break translation symmetry have uniform, or flat, momentum distributions, and delocalized states may have single peaks with finite-width or distributions of peaks (see Section \ref{['sec: AA model']}). Upon twisting boundary conditions, the LRE/delocalized states have their momentum-distribution peaks shifted, and can be distinguished from the un-shifted distributions. In contrast, for SRE/localized states the distributions are shifted by twisting boundary conditions, but are not distinguishable from the un-shifted cases, and the many-body momentum is unchanged.
• Figure 2: Localization phase diagrams for the ground state of the DDM at different chemical potentials $\mu$ and momentum coupling strengths $V/t$. Here, the total system size is $L=900a$, and $\Delta=48\frac{2\pi}{L}$. The dark-blue region in (a), for $z_x\to0$, and the bright-yellow region in (b), for $z_t\to1$, correspond to the completely delocalized phase of the DDM, and the other regions are partially localized.
• Figure 3: Position and momentum distributions for the ground state of the DDM at $\mu=0.0$. The blue $V/t=0.01$ curve is completely delocalized (flat) in position space (a), while sharply peaked and completely localized in momentum space (b).The orange $V/t=0.25$ curve is partially localized in position space (a), and completely delocalized (flat) in momentum space (b).
• Figure 4: Position (blue) and momentum (orange) distributions for single-particle eigenstates of the DDM at $V/t=0.021$. (a) Eigenstate of the type $|\psi_{(k)}\rangle$ at energy $E = 2.99$, whose degeneracy was not lifted by the perturbation. (b)Eigenstate of the type $|\tilde{\psi}_{|k|}\rangle$ at energy $E=-5.13$, whose degeneracy was lifted by the perturbation.
• Figure 5: Localization measures for each single-particle state, arranged by energy, at different momentum hopping strengths $V/t$, for $L=900a$. (a) Single-particle $z_x$ for the eigenstates of the DDM. Each value of $V/t$ is plotted with a negative vertical offset for clarity. (b) Single-particle $z_t$ for the eigenstates of the DDM. (c) Comparison between the single-particle $z_x$, $z_t$, and the localization lengths in position, $\lambda_x$, and momentum space, $\lambda_k$, for $V/t=0.015$.