Entanglement spectrum of a topological phase in one dimension (0910.1811v2)

Abstract: We show that the Haldane phase of S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\pi$-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, "topologically trivial", phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

• The paper establishes that double degeneracy in the entanglement spectrum signifies the Haldane phase under symmetry protection.
• It employs matrix product states to show how projective representations of symmetries classify topologically non-trivial phases.
• The findings offer experimental benchmarks and frameworks for investigating and controlling quantum phases in low-dimensional systems.

Entanglement Spectrum of a Topological Phase in One Dimension

The paper "Entanglement spectrum of a topological phase in one dimension" by PoLLMann et al. advances our understanding of symmetry-protected topological (SPT) phases, particularly focusing on the well-known Haldane phase of spin-1 chains. This work provides a framework for classifying gapped phases in one-dimensional systems by examining the entanglement spectrum, presenting a robust conceptual tool that potentially expands existing paradigms in condensed matter physics.

Key Findings and Discussions

1. Entanglement Spectrum Degeneracy: The authors show that the Haldane phase is characterized by a ubiquitous double degeneracy in the entanglement spectrum, protected by specific symmetries such as time-reversal, bond-centered inversion, or a dihedral group of π-rotations about two orthogonal axes. This degeneracy stands unless the system transitions into a topologically trivial phase or these symmetries are disrupted.
2. Symmetry Protection: The stability of the Haldane phase, even in settings devoid of conventional symmetry-breaking order parameters, is elucidated through symmetry protection. The preservation of symmetries like time-reversal or inversion ensures the persistence of the Haldane phase against perturbations that would generally destabilize other phases.
3. Matrix Product States (MPS) and Projective Representations: A methodological contribution lies in employing MPS to analyze how sacred symmetries manifest in the transformation properties of ground state wavefunctions. The transformation of Schmidt eigenstates under these symmetries involves projective representations, providing a framework for identifying and classifying topologically non-trivial phases.
4. Modified Symmetries and Generalizations: The paper extends the notion of symmetry protection to accommodate scenarios where traditional symmetries are broken. For instance, in systems with Dzyaloshinskii-Moriya interactions, a modified inversion symmetry could protect a Haldane-like phase, allowing the framework to adapt to various physical situations.
5. Experimental Consequences: Theoretical predictions suggest practical implications, such as maintaining a minimum entanglement entropy of $\ln(2)$ during adiabatic bond weakening in the Haldane phase. This perspective opens pathways for experimental verification using concepts like relevant correlation functions and residual entanglement.

Implications and Future Directions

The work by PoLLMann et al. offers significant insights into the nature of SPT phases and outlines a systematic approach to their classification. By establishing entanglement spectrum degeneracy as a haLLMark of topological phases, this paper proposes a unifying criterion that could assist in identifying and distinguishing various gapped phases.

From a practical standpoint, the implications of these findings extend to designing and interpreting experiments involving quantum entanglement and SPT phases in low-dimensional systems. The potential to classify phases based on symmetry and entanglement characteristics might impact the development of quantum materials and technologies, where precise control over quantum states is crucial.

Future research could build upon this foundation to explore higher-dimensional counterparts of these concepts, investigate the interplay with thermal fluctuations, and delve into the computational aspects underlying this classification scheme. As the landscape of quantum materials broadens, the theoretical constructs discussed in this paper are anticipated to serve as pivotal benchmarks in the ongoing quest for understanding complex quantum topological matter.