When Is Kramers-Wannier Duality Invertible?
Abstract
{\it Kramers-Wannier} duality of the quantum Ising chain is naturally realized as a noninvertible transformation on a finite periodic chain. Upon incorporating appropriate symmetry-twisted sectors, however, the duality can be promoted to an invertible unitary transformation. In this Letter, we classify when the Kramers-Wannier duality admits an invertible realization on a finite Hilbert space. We show that this is determined by the representation of the Ising bond algebra and establish a necessary and sufficient condition in terms of its two central elements. This representation-theoretic criterion unifies the conventional noninvertible realization with invertible constructions involving symmetry-twisted sectors and applies equally to other models with the same underlying bond algebra. As an explicit example, we find an order-disorder duality in a non-trivial realization of the Ising bond algebra that admits an invertible Kramers-Wannier duality.
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BibTeX
@misc{sinha2026when,
title = {{When Is Kramers-Wannier Duality Invertible?}},
author = {Akash Sinha and Pramod Padmanabhan and Vladimir Korepin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20090},
eprint = {2609.20090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20090},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - When Is Kramers-Wannier Duality Invertible?
AU - Akash Sinha
AU - Pramod Padmanabhan
AU - Vladimir Korepin
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.20090
UR - https://arxiv.org/abs/2609.20090
AB - {\it Kramers-Wannier} duality of the quantum Ising chain is naturally realized as a noninvertible transformation on a finite periodic chain. Upon incorporating appropriate symmetry-twisted sectors, however, the duality can be promoted to an invertible unitary transformation. In this Letter, we classify when the Kramers-Wannier duality admits an invertible realization on a finite Hilbert space. We show that this is determined by the representation of the Ising bond algebra and establish a necessary and sufficient condition in terms of its two central elements. This representation-theoretic criterion unifies the conventional noninvertible realization with invertible constructions involving symmetry-twisted sectors and applies equally to other models with the same underlying bond algebra. As an explicit example, we find an order-disorder duality in a non-trivial realization of the Ising bond algebra that admits an invertible Kramers-Wannier duality.
ER -