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

TL;DR

This paper proves that long-range interacting quantum spin chains with certain decay conditions exhibit entanglement area laws and are constrained by Lieb-Schultz-Mattis theorems, revealing fundamental links between symmetry, entanglement, and long-range interactions.

Contribution

It establishes the entanglement area law and Lieb-Schultz-Mattis constraints for long-range systems with specific decay rates and symmetries, extending known results to more general long-range interactions.

Findings

Ground states satisfy the entanglement area law under decay faster than 1/r^d for gapped systems.

Long-range Hamiltonians with anomalous symmetry cannot have a unique gapped symmetric ground state.

Pure states with anomalous symmetry are necessarily long-range entangled.

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.