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Symmetry protection of topological order in one-dimensional quantum spin systems

Frank Pollmann, Erez Berg, Ari M. Turner, Masaki Oshikawa

TL;DR

This work investigates symmetry-protected topological order in one-dimensional integer-spin chains, focusing on the Haldane phase and the symmetries required for its protection. Using Kennedy–Tasaki hidden symmetry arguments, edge-state considerations, and matrix-product-state formalism, it shows that odd-$S$ Haldane phases are robust as long as any of D$_2$, time-reversal, or inversion symmetry is preserved, while even-$S$ AKLT states are not protected and can be adiabatically connected to trivial states. It provides explicit symmetry-preserving constructions and numerical evidence demonstrating the triviality of even-$S$ AKLT states and the adiabatic connections to dimerized or rung-singlet phases. The results clarify the criteria for symmetry-protected topological order in 1D spin systems and extend to spin ladders, offering a practical framework for classifying and identifying SPT phases in 1D materials and models.

Abstract

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $π$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \rightarrow - S^{x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \textbf{81}, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

Symmetry protection of topological order in one-dimensional quantum spin systems

TL;DR

This work investigates symmetry-protected topological order in one-dimensional integer-spin chains, focusing on the Haldane phase and the symmetries required for its protection. Using Kennedy–Tasaki hidden symmetry arguments, edge-state considerations, and matrix-product-state formalism, it shows that odd- Haldane phases are robust as long as any of D, time-reversal, or inversion symmetry is preserved, while even- AKLT states are not protected and can be adiabatically connected to trivial states. It provides explicit symmetry-preserving constructions and numerical evidence demonstrating the triviality of even- AKLT states and the adiabatic connections to dimerized or rung-singlet phases. The results clarify the criteria for symmetry-protected topological order in 1D spin systems and extend to spin ladders, offering a practical framework for classifying and identifying SPT phases in 1D materials and models.

Abstract

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd- Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of -rotations about and axes; (ii) time-reversal symmetry ; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \textbf{81}, 064439 (2010)]. On the other hand, an even- Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

Paper Structure

This paper contains 12 sections, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Haldane phase in the presence of global D$_2$ symmetry
  3. Stability of the Haldane phase
  4. Characterization by edge states in the presence ot Time reversal symmetry
  5. Inversion symmetry of a ring
  6. Argument based on the Matrix-Product state representation
  7. Triviality of the Even $S$ AKLT state
  8. From $S=2$ AKLT to large-$D$ limit
  9. From the AKLT state to a dimerized phase
  10. Spin ladders
  11. Conclusions
  12. MPS path from a dimerized state to the AKLT state

Figures (6)

  • Figure 1: Four symmetry broken states (upper panel) which are obtained by applying the non-local transformation $U_{KT}$ to the degenerate edge states (lower panel). Note that the arrows in the upper panel represent the spin polarization in the bulk, while in the lower panel they represent the spin polarization at the edges.
  • Figure 2: (Color online) The $S=1$ AKLT state on a ring with $L=7$ sites. The connecting lines represent spin-$\frac{1}{2}$ singlets. We consider the lattice inversion about the vertical line.
  • Figure 3: (Color online) Eigenvalue spectrum of the transfer matrix (completely positive map \ref{['eq_CPM']}) for the interpolating MPS defined in \ref{['eq_interpolated']}. Except for the largest eigenvalue (unity), all the eigenvalues have absolute value smaller than $1$ for $0 \leq t \leq 1$, implying finite correlation length. Thus the trivial state $|\mathcal{D}\rangle$ at $t=0$ and the $S=2$ AKLT state at $t=1$ are adiabatically connected without any phase transition.
  • Figure 4: (Color online) (a) Dimerized state of spin-$S$ singlets on every second bond at $t=0$ and the AKLT state formed by $S/2$ singlets on every bond at $t=1$. (b) Eigenvalue spectrum of the two-site transfer matrix along the path connecting the fully dimerized state and the AKLT state for the spin $S=2$ chain (see text for details).
  • Figure 5: (Color online) (a) Ladder geometry used for the calculation. Entanglement spectra for (b) $S=\frac{1}{2}$ and (c) $S=1$ ladders. The entanglement spectrum is plotted versus the ratio $R=J_{\text{rung}}/(J_{\text{leg}}+|J_{\text{rung}}|)$ for $J_{\text{leg}}>0$. Thus $R=-1$ corresponds to infinite ferromagnet coupling on the rungs and $R=1$ to infinite antiferromagnet couplings. The number of dots on each level indicates its degeneracy.
  • ...and 1 more figures