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Generalized Kramers-Wanier Duality from Bilinear Phase Map.

Linhao Li^1,2, Masaki Oshikawa^3,4, Han Yan³

¹Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.

Physical Review Letters

|July 7, 2026

Summary

Researchers developed the bilinear phase map to study unconventional quantum phases in qudit spin chains. This new method generalizes existing transformations, revealing new dualities and anomaly classifications for lattice models.

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Area of Science:

Condensed Matter Physics
Quantum Information Theory
High Energy Physics

Background:

The Kramers-Wannier transformation is a key tool for understanding gapped phases in quantum systems.
Investigating unconventional gapped phases in qudit (multi-level quantum system) spin chains requires advanced theoretical frameworks.
Generalized quantum phases and dualities, including non-unitary transformations, present significant theoretical challenges.

Purpose of the Study:

To introduce and explore the bilinear phase map as a generalization of the Kramers-Wannier transformation.
To investigate unconventional gapped phases in qudit spin chains.
To derive general noninvertible fusion rules and constrain anomaly conditions in lattice duality models.

Main Methods:

Developed the bilinear phase map by encoding the generalized Kramers-Wannier transformation in a matrix.
Applied the bilinear phase map to explore a broad spectrum of quantum phases and dualities.
Derived general noninvertible fusion rules.
Obtained strong constraints on anomaly conditions for qudit models with duality symmetries.

Main Results:

The bilinear phase map successfully generalizes the Kramers-Wannier transformation for studying unconventional qudit spin chain phases.
Exploration revealed a broader range of generalized quantum phases and dualities, including non-unitary aspects.
General noninvertible fusion rules were derived.
Significant constraints on anomaly conditions for qudit models with duality symmetries were established.

Conclusions:

The bilinear phase map provides a powerful new tool for investigating complex quantum phases and dualities.
This framework leads to a microscopic anomaly classification for a wide array of lattice dualities.
The study advances the understanding of topological phases and symmetries in quantum many-body systems.