Quantized six-vertex model at roots of unity: Frobenius property and free-parafermion spectra

Abstract

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at q=ε, where ε is a primitive Nth root of unity with N odd. For the commuting layer transfer matrices 𝕋(y) with mixed boundary conditions, we establish the quantum Frobenius property 𝕋(y)𝕋(ε y)⋯ 𝕋(ε^N−1y) = 𝒫_N(y^N)𝕀, where 𝒫_N is a scalar Laurent polynomial obtained directly from the monomial expansion of 𝕋(y). Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting τ₂ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

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BibTeX RIS

BibTeX

@misc{inoue2026quantized,
  title = {{Quantized six-vertex model at roots of unity: Frobenius property and free-parafermion spectra}},
  author = {Rei Inoue and Atsuo Kuniba and Yuji Terashima and Junya Yagi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27740},
  eprint = {2609.27740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27740},
}

RIS

TY  - UNPB
TI  - Quantized six-vertex model at roots of unity: Frobenius property and free-parafermion spectra
AU  - Rei Inoue
AU  - Atsuo Kuniba
AU  - Yuji Terashima
AU  - Junya Yagi
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27740
UR  - https://arxiv.org/abs/2609.27740
AB  - We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at q=ε, where ε is a primitive Nth root of unity with N odd. For the commuting layer transfer matrices 𝕋(y) with mixed boundary conditions, we establish the quantum Frobenius property 𝕋(y)𝕋(ε y)⋯ 𝕋(ε^N−1y) = 𝒫_N(y^N)𝕀, where 𝒫_N is a scalar Laurent polynomial obtained directly from the monomial expansion of 𝕋(y). Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting τ_ 2 model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.
ER  -

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