Lattice non-invertible symmetry from non-commuting transfer matrices
Abstract
Conventional quantum integrability is encoded in a commuting algebra of transfer matrices. By contrast, several models possess additional non-commuting conserved charges with important physical consequences, yet the nature of the corresponding symmetry has remained elusive. Focusing on the XXZ spin chain at roots of unity, we show that the non-Abelian analogue of the commuting transfer-matrix algebra is governed by quadratic relations following from a new class of unbalanced Yang-Baxter/RLL relations. This quadratic algebra is shown to encode precisely the Onsager algebra, for which we construct explicit matrix-product representations of both its generators and its duality defect line. The latter obeys ZN Tambara-Yamagami fusion rules, thereby providing a lattice realization of the topological defect lines of the compactified boson conformal field theory. Our results identify non-Abelian transfer-matrix algebras as a microscopic origin for Onsager symmetry and dualities in lattice models.
Related
- Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices
- Symmetry-permuting entanglers for non-invertible symmetry-protected topological phases
- Neutrino Phenomenology and Leptogenesis from Non-Invertible Fusion Symmetry
- Non-invertible symmetries and modular invariance in lattice models
- On general background of quantum non-invertible symmetry in 2D
BibTeX
@misc{vernier2026lattice,
title = {{Lattice non-invertible symmetry from non-commuting transfer matrices}},
author = {Eric Vernier and Yuan Miao and Masahito Yamazaki},
howpublished = {HAL},
year = {2026},
month = jul,
url = {https://hal.science/hal-05846587v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lattice non-invertible symmetry from non-commuting transfer matrices
AU - Eric Vernier
AU - Yuan Miao
AU - Masahito Yamazaki
PY - 2026
DA - 2026/07/22
PB - HAL
UR - https://hal.science/hal-05846587v1
AB - Conventional quantum integrability is encoded in a commuting algebra of transfer matrices. By contrast, several models possess additional non-commuting conserved charges with important physical consequences, yet the nature of the corresponding symmetry has remained elusive. Focusing on the XXZ spin chain at roots of unity, we show that the non-Abelian analogue of the commuting transfer-matrix algebra is governed by quadratic relations following from a new class of unbalanced Yang-Baxter/RLL relations. This quadratic algebra is shown to encode precisely the Onsager algebra, for which we construct explicit matrix-product representations of both its generators and its duality defect line. The latter obeys ZN Tambara-Yamagami fusion rules, thereby providing a lattice realization of the topological defect lines of the compactified boson conformal field theory. Our results identify non-Abelian transfer-matrix algebras as a microscopic origin for Onsager symmetry and dualities in lattice models.
ER -