Symmetry protection of topological order in one-dimensional quantum spin systems (0909.4059v3)

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.

• The paper shows that global symmetries, such as dihedral, time-reversal, and inversion, protect the topologically non-trivial odd-S Haldane phase.
• The paper reveals that applying non-local unitary transformations exposes hidden Z2×Z2 symmetry breaking tied to edge states in 1D systems.
• The paper supports its claims with robust numerical evidence and Matrix Product State analysis, indicating gap closure or symmetry change during transitions.

Symmetry Protection of Topological Phases in One-Dimensional Quantum Spin Systems

The paper by PoLLMann et al. provides an in-depth analysis of the topological properties of the Haldane phase in integer spin chains. It fundamentally investigates the interplay between symmetry and topological order, demonstrating how certain symmetries can stabilize topological phases in one-dimensional quantum systems. This work is crucial for understanding the broader implications of topological phases, particularly in condensed matter physics and the development of quantum materials.

Key Findings

The authors focus on the properties of the well-known Haldane phase, which emerges in integer spin chains and was initially described by Haldane's conjecture. According to this hypothesis, Heisenberg antiferromagnetic chains with integer spins exhibit a gap in their excitation spectrum, contrasting with their half-integer counterparts, which are gapless.

1. Symmetry Protection: The paper articulates that an odd-S Haldane phase is topologically non-trivial and is safeguarded by any of the following global symmetries:
   • The dihedral group of π-rotations around the x, y, and z axes.
   • Time-reversal symmetry.
   • Link inversion symmetry. In contrast, even-S Haldane phases do not enjoy such protection and are thus indistinguishable from trivial, site-factorizable phases.
2. Edge States and Hidden Symmetry: The analysis reveals that in odd-S chains, the presence of edge states can be tied to a hidden $Z_2 \times Z_2$ symmetry breaking, which becomes manifest upon applying non-local unitary transformations. This framework extends our understanding of how topological characteristics in lower dimensions can emerge without the presence of a local order parameter.
3. Numerical Evidence and Matrix Product State (MPS) Approach: The authors provide rigorous numerical evidence supported by MPS analysis, emphasizing the robustness of their theoretical claims. They exploit the properties of MPS to demonstrate that for odd-S systems, transitions to trivial phases must entail closing of a gap or a significant change in symmetry properties.

Implications and Future Directions

The results shed light on the nature of topological order and its dependence on symmetry in one-dimensional systems. The recognition that symmetry protection might vary between odd and even spin chains opens up new avenues for exploring analogous phenomena in higher dimensions or in systems with different physical constraints.

• Practical Implications: Understanding symmetry protection mechanisms is essential for the potential realization of quantum technologies, where stability against perturbations and decoherence induced by symmetry-preserving interactions can be leveraged.
• Theoretical Developments: The methodology and approaches detailed can inspire future studies in examining the role of symmetries in even more complex systems, including those involving higher-spin excitations or multi-leg spin ladders.

Overall, the paper robustly contributes to the discourse on topological quantum phases and sets the stage for ongoing explorations into the exotic nature of quantum matter under various symmetry constraints.