Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement

TL;DR

The paper advances a unified framework showing that in 1D long-range interacting systems, gapped admissible Hamiltonians obey an entanglement area law even amid ground-state degeneracy, enabling a generalized LSM theorem for anomalous symmetries. By linking anomaly indices (via QCAs) to clustering and area-law properties, it proves that anomalous symmetry forbids symmetric locally unique gapped ground states and enforces long-range entanglement in symmetric pure states. It further shows that symmetry-enforced long-range entanglement persists across finite systems and can be diagnosed with Ohya-Petz entropy and von Neumann algebra techniques. The results apply to multi-body, non-on-site, and translation-symmetric settings and have broad relevance for platforms with long-range interactions, such as trapped ions and Rydberg arrays.

Abstract

We establish multiple interrelated, fundamental results in quantum many-body systems that can have long-range interactions. For a sufficiently long quantum spin chain, we first show that if the multi-spin interactions in the Hamiltonian decay fast enough as their ranges increase and the Hamiltonian is gapped, then the ground states satisfy the entanglement area law, even if there is a ground state degeneracy due to a spontaneously broken discrete symmetry. This area law also holds for certain excited states. Second, if such a long-range interacting Hamiltonian has an anomalous symmetry, then the Lieb-Schultz-Mattis theorem applies, i.e., the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, these results hold when the interactions decay faster than $1/r^2$, with $r$ the distance between the two interacting spins. Third, we show that pure states with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of these results through various examples.

Figures (6)

• Figure 1: An illustration on how to obtain anomaly index $\omega\in\mathrm{H}^{3}(G;\mathrm{U}(1))$ from the symmetry action $\alpha$.
• Figure 2: The interaction of two disjoint intervals $X,Y$ seprated by distance $d$. The range of interaction is denoted by $Z$ with $\mathrm{diam}(Z)=r$.
• Figure 3: The flow chart for proving the entanglement area law. Some of the results are stated informally above, and the corresponding precise statements can be found in the paper.
• Figure 4: The original lattice $\mathbb{Z}$ is regrouped into $(2q+3)$ blocks. The case $q=4$ is shown in the figure. The length of blue blocks is $l$ while the length of pink blocks is infinite or depends on $n$.
• Figure 5: The whole chain is folded with respect to the midpoint of the interval $[0,n]$.

Theorems & Definitions (281)

• Theorem 4.1: Theorem \ref{['thm:area_law']} in Appendix \ref{['subapp: proving area law']}
• Proposition 4.1: Informal
• Lemma 5.1
• Theorem 5.1
• Theorem 5.2: Theorem \ref{['thm:limit_ad_gs']} in Appendix \ref{['sec: thermodynamic limit']}
• Corollary 5.1
• Theorem 6.1: Corollary \ref{['coro:SRE']} in Appendix \ref{['subsec: thermodynamic limit of SRE']}
• Theorem 6.1: Corollary \ref{['coro:SRE']} in Appendix \ref{['subsec: thermodynamic limit of SRE']}
• Theorem 6.2: Theorem \ref{['thm:finite_SRE']} in Appendix \ref{['subsec: thermodynamic limit of SRE']}
• Definition A.1