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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.

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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.