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

TL;DR

This paper analyzes the topological protection of the Haldane phase in one-dimensional integer spin chains, showing that odd spins have symmetry-protected topological order, while even spins do not.

Contribution

It demonstrates that the Haldane phase's topological protection depends on specific symmetries and distinguishes between odd and even spin chains.

Findings

Odd-S Haldane phase is topologically non-trivial and symmetry-protected.

Even-S Haldane phase is not topologically protected.

Numerical evidence supports the symmetry-based protection claims.

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 $\pi$-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.