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

R. Liu, Jinmin Yi+2 more|arXiv (Cornell University)|May 23, 2024

Complex Systems and Time Series AnalysisEconomics, Econometrics and Finance3 citations

TL;DR

This paper establishes generalized Lieb-Schultz-Mattis theorems for quantum spin chains with long-range interactions, proving that systems with fast-decaying interactions ($1/r^{rak{a}}$, $rak{a}>2$) and anomalous symmetries cannot have a unique gapped symmetric ground state. It further shows that any pure symmetric state in such systems must be long-range entangled, extending LSM constraints beyond ground states and to broader symmetry classes including continuous and non-on-site internal symmetries.

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.

Motivation & Objective

• To extend Lieb-Schultz-Mattis theorems to quantum spin chains with long-range interactions, particularly those decaying as $1/r^{rak{a}}$ with $rak{a}>2$.
• To establish that systems with anomalous symmetries—such as on-site internal symmetries combined with lattice translation or purely non-on-site internal symmetries—cannot support a unique gapped symmetric ground state.
• To generalize the theorems beyond ground states to include all pure symmetric states, showing they must be long-range entangled.
• To unify and extend previous LSM-type constraints by incorporating both long-range interactions and broader symmetry structures, including continuous and discrete symmetries.
• To provide a framework applicable to realistic models such as long-range Ising and dipolar spin models, with implications for phase classification and topological order.

Proposed method

• Employing operator algebra formalism for infinite systems to define locality and symmetry actions, enabling rigorous treatment of long-range interactions.
• Using the anomaly index from group cohomology $H^3(G;U(1))$ to classify and characterize anomalous symmetries, particularly in the context of $G = O(2) imes bZ$ or $bZ_2$.
• Deriving a sufficient condition for the absence of a unique gapped symmetric ground state via the admissibility condition on interaction decay, specifically requiring $rak{a}>2$ for 2-body interactions.
• Applying the formalism to finite systems by taking thermodynamic limits, ensuring results are physically meaningful for realistic systems.
• Constructing explicit models such as the long-range $O(2) imesbZ$ symmetric model and the non-local $bZ_2$ symmetric Hamiltonian with $J_{ij} o |i-j|^{-rak{a}}$, $rak{a}>2$, to demonstrate the theorems.
• Using the symmetry anomaly and entanglement structure to prove that all symmetric pure states must be long-range entangled, regardless of whether they are ground states.

Experimental results

Research questions

• RQ1 Can Lieb-Schultz-Mattis-type theorems be generalized to quantum spin chains with long-range interactions that decay as $1/r^{rak{a}}$?
• RQ2 Under what conditions on interaction decay and symmetry structure does a unique gapped symmetric ground state become impossible?
• RQ3 Do the LSM constraints extend to all pure symmetric states, not just ground states, and do they enforce long-range entanglement?
• RQ4 Can the theorems be applied to systems with non-on-site internal symmetries or continuous symmetries, such as $O(2)$?
• RQ5 How do the results apply to realistic long-range models such as dipolar or Rydberg atom chains with tunable interaction exponents?

Key findings

• For 2-body long-range interactions decaying as $1/r^{rak{a}}$, the LSM theorem holds when $rak{a}>2$, ruling out a unique gapped symmetric ground state if the system has an anomalous symmetry.
• Any pure symmetric state in such systems—whether a ground state or not—must be long-range entangled, a direct consequence of the symmetry anomaly and interaction decay.
• The theorems apply to a broad class of symmetries, including on-site internal symmetries combined with lattice translation, purely internal non-on-site symmetries, and both discrete and continuous groups.
• The model with $O(2) imesbZ$ symmetry and long-range $1/r^{rak{a}}$ interactions ($rak{a}>2$) cannot have a unique gapped symmetric ground state, consistent with known phase diagrams showing only gapless or broken-symmetry phases.
• The long-range $bZ_2$ symmetric Hamiltonian with $J_{ij} o |i-j|^{-rak{a}}$, $rak{a}>2$, is either gapless or spontaneously breaks $bZ_2$ symmetry, in agreement with the theorems.
• The results are robust to explicit breaking of translation symmetry, as long as the interaction decay and symmetry anomaly conditions are preserved, demonstrating the central role of the anomaly and decay rate.