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Cluster State as a Noninvertible Symmetry-Protected Topological Phase
Sahand Seifnashri1, Shu-Heng Shao2
1School of Natural Sciences, <a href="https://ror.org/00f809463">Institute for Advanced Study</a>, Princeton, New Jersey, USA.
Physical Review Letters
|September 27, 2024
Summary
The standard cluster model exhibits noninvertible symmetry, classifying it as a non-invertible symmetry-protected topological (SPT) phase. This discovery reveals new insights into topological phases and their unique properties.
Area of Science:
- Condensed Matter Physics
- Quantum Information Theory
- High Energy Physics
Background:
- Symmetry-protected topological (SPT) phases are crucial in understanding quantum matter.
- The standard 1+1D Z_{2}×Z_{2} cluster model is a well-studied example of an SPT phase.
- Noninvertible symmetries represent a novel extension beyond conventional symmetries in topological phases.
Purpose of the Study:
- To investigate the global symmetry of the 1+1D Z_{2}×Z_{2} cluster model.
- To classify the cluster state as a non-invertible symmetry-protected topological (SPT) phase.
- To identify and characterize new Rep(D_{8}) SPT phases and their properties.
Main Methods:
- Analysis of the 1+1D Z_{2}×Z_{2} cluster model to determine its global symmetry.
- Characterization of the symmetry using the fusion category Rep(D_{8}).
- Construction of commuting Pauli Hamiltonians for new Rep(D_{8}) SPT phases in a qubit system.
Main Results:
- The 1+1D Z_{2}×Z_{2} cluster model possesses a noninvertible global symmetry, Rep(D_{8}).
- The cluster state is identified as both a Z_{2}×Z_{2} and a non-invertible SPT phase.
- Two new commuting Pauli Hamiltonians for Rep(D_{8}) SPT phases were discovered, consistent with theoretical classifications.
- Edge modes and local projective algebras at interfaces between these phases were identified.
- The absence of a symmetric entangler between distinct SPT states was demonstrated.
Conclusions:
- The study establishes the 1+1D Z_{2}×Z_{2} cluster model as a non-invertible SPT phase.
- New insights into the classification and properties of non-invertible SPT phases are provided.
- The findings bridge theoretical classifications in field theory and mathematics with concrete physical realizations in qubit systems.
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