[Paper Review] Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries

Yuan Yao, Linhao Li+2 more|arXiv (Cornell University)|Jul 19, 2023

Physics of Superconductivity and Magnetism8 citations

TL;DR

The paper extends LSM-type ingappability to 1D quantum magnets with antiunitary translation or inversion symmetries, showing half-integer spin chains must be gapless or degenerate under these symmetries, even with limited spin-rotation symmetry.

ABSTRACT

We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete $\mathbb{Z}_2 imes\mathbb{Z}_2$ global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.

Motivation & Objective

• Motivate and establish ingappability constraints for 1D quantum magnets when antiunitary lattice symmetries are present.
• Generalize LSM-type results beyond conventional spin-rotation and translation symmetries to include antiunitary translation and inversion.
• Demonstrate robustness of spectrum under symmetry twisting and connect to SPT bulk-boundary correspondence and lattice homotopy.

Proposed method

• Introduce a chiral spin chain with a staggered triple-product interaction that preserves antiunitary lattice symmetries.
• Use symmetry-twisting (STBC) and spectrum-robustness arguments to relate twisted and untwisted Hamiltonians.
• Show that antiunitary translation/inversion lead to exact degeneracies in the twisted spectrum, implying ingappability for half-integer spins under PBC.
• Apply a gauge-transformation perspective to define modified antiunitary symmetries and compute commutation relations that trigger degeneracies.
• Connect field-theoretic anomalies in the low-energy theory to lattice ingappability via bosonization and anomalous textures.
• Present a lattice-homotopy framework that classifies symmetry generators yielding LSM-type ingappability.

Experimental results

Research questions

• RQ1 Under what antiunitary lattice symmetries do 1D quantum magnets with half-integer spins become ingappable?
• RQ2 Can antiunitary translation or inversion symmetries guarantee gaplessness or ground-state degeneracy beyond conventional LSM constraints?
• RQ3 How do anomalies in low-energy field theories relate to lattice ingappability in chiral spin models?
• RQ4 What minimal symmetry generators are sufficient to enforce LSM-type ingappability in 1D systems with various spin interactions?
• RQ5 How does lattice homotopy classify these ingappability constraints?

Key findings

• Half-integer spin chains with antiunitary translation or inversion symmetries and discrete spin-rotation symmetry are either gapless or have degenerate ground states under periodic boundary conditions.
• Ingappability persists even when spin-rotation symmetry is explicitly broken, provided antiunitary inversion symmetry I2 is preserved.
• Antiunitary translation T2 alone does not guarantee ingappability; I2 alone can enforce it, interpreted as a boundary of a 2D I2-protected topological phase.
• Twisted-boundary analysis reveals exact degeneracies in H_tw due to antiunitary symmetry commutation relations, implying ground-state degeneracy under PBC if s is half-integer.
• Anomalies in the low-energy (bosonized) theory (e.g., mixed anomalies with rotation symmetries and ’t Hooft anomalies for I2) align with the lattice ingappability conclusions.
• Table 1 (symmetry classes) provides minimal generator sets that ensure LSM-type ingappability for 1D SU(2) spins across various spin interactions.