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Identification of gapless phases by a twisting operator

Hang Su, Tengzhou Zhang, Yuan Yao, Akira Furusaki

TL;DR

The paper establishes a rigorous, general necessary condition for 1D spin chains with SO(3) symmetry to be gapped, showing that ground states must be SU(2) singlets and that the ground-state expectation $\langle\hat{U}^2\rangle$ approaches unity with $\mathscr{O}(1/L)$ corrections in the thermodynamic limit. By proving the $\langle\hat{U}^2\rangle$-criterion and demonstrating its effectiveness on diverse models (MG, HAF, BLBQ, AKLT), the authors provide a practical diagnostic to identify gapless phases and supply evidence for gapped ones. The approach complements LSM-type results, offering a numerically tractable test that does not require stringent microscopic unit-cell constraints. The findings have broad implications for classifying quantum spin chains and may extend to open boundaries and fermionic systems via related mappings.

Abstract

We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases, and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.

Identification of gapless phases by a twisting operator

TL;DR

The paper establishes a rigorous, general necessary condition for 1D spin chains with SO(3) symmetry to be gapped, showing that ground states must be SU(2) singlets and that the ground-state expectation approaches unity with corrections in the thermodynamic limit. By proving the -criterion and demonstrating its effectiveness on diverse models (MG, HAF, BLBQ, AKLT), the authors provide a practical diagnostic to identify gapless phases and supply evidence for gapped ones. The approach complements LSM-type results, offering a numerically tractable test that does not require stringent microscopic unit-cell constraints. The findings have broad implications for classifying quantum spin chains and may extend to open boundaries and fermionic systems via related mappings.

Abstract

We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases, and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.

Paper Structure

This paper contains 16 sections, 53 equations, 3 figures.

Table of Contents

  1. Introduction.---
  2. Symmetry, locality, and extended translation.---
  3. Lemma 1.---
  4. Corollary 4.---
  5. Theorem 5: $\langle\hat{U}^2\rangle$-criterion.---
  6. Identifying gaplessness and evidence of gappedness.---
  7. Conclusions and discussions.---
  8. Acknowledgements.---
  9. Data availablity---
  10. Supplemental Materials
  11. Analytical results of for Majumdar-Ghosh model defined on even-length chains
  12. Periodic boundary conditions (PBCs) results
  13. Open Boundary Conditions (OBCs) results
  14. Analytical results of for Affleck-Kennedy-Lieb-Tasaki model
  15. PBCs results
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Results for $\langle\hat{U}^2\rangle$ in the gapless spin-1/2 models: Majumdar-Ghosh model with odd chain lengths (left) and Heisenberg antiferromagnetic chain (right). Both show the tendency of $\langle\hat{U}^2\rangle \ll 1$. The latter dataset is fitted by $\langle\hat{U}^2\rangle = a - b/\ln L$, with $a=-0.009(4)$, $b=0.346(7)$, shown in the inset.
  • Figure 2: Results for $\langle\hat{U}^2\rangle$ as a function of $1/L$ for the BLBQ chains. The left is $\theta=0$ case, i.e., spin-1 HAF chain, which is fitted by Eq. \ref{['DMRG_fitting']}, yielding $a=1.000(1)$ and $b = 22.927(7)$, $k = 0.984(9)$. The right one is two gapless points, including QMC results for $\theta=-3\pi/4$ case and DMRG results for $\theta=-\pi/4$ case are shown, resepectviely.
  • Figure 3: Results for $\langle\hat{U}^2\rangle$ as a function of $1/L$ for spin-$1/2$ (left) and spin-$1$ (right) HAF chains, with both OBCs and PBCs.