The Yang-Baxter integrability of the critical Ising chain
Akash Sinha, Tinu Justin, Pramod Padmanabhan, Vladimir Korepin
TL;DR
The authors establish Yang-Baxter integrability for the 1D critical transverse-field Ising model by constructing a Majorana-fermion $R$-matrix that satisfies the Yang-Baxter equation with additive spectral parameters, enabling a commuting transfer-matrix structure despite non-locality and non-regularity. They derive the associated local Hamiltonian via the boost-operator method and show that, after Jordan-Wigner transformation, it corresponds to the critical TFIM with a parity twist, connecting to the Kitaev chain at criticality in the thermodynamic limit. The work further reveals non-invertible Kramers-Wannier duality as a symmetry arising from transfer-matrix constructions, and it systematically generates higher conserved charges ${oldsymbol{ ext Q}}_p={ m i} sum_j oldsymbol{ mgamma_j}oldsymbol{ mgamma_{j+p-1}}$, highlighting how non-local R-matrices fit within the quantum inverse scattering framework. These results broaden the scope of QISM to non-local, non-regular $R$-matrices and establish deep links between integrability, duality symmetries, and topological features of the critical Ising/Kitaev system, with potential implications for exact results and quantum simulations. Future directions include developing an algebraic Bethe Ansatz for these new $R$-matrices and exploring their use in integrable quantum circuits and non-invertible symmetry frameworks.
Abstract
We show that the one dimensional, critical transverse field Ising model is Yang-Baxter integrable. This is done by constructing commuting transfer matrices built out of a $R$-matrix satisfying the Yang-Baxter equation with additive spectral parameters. The $R$-matrix is non-local, as it is expressed in terms of Majorana fermions. It is also non-regular. Nevertheless, we show that the quantum inverse scattering method can still be suitably adapted. We then recursively obtain the conserved quantities [in the infinite volume] by the boost operator method. Remarkably, among the conserved charges we also find the Kramers-Wannier duality and other non-invertible symmetries for the periodic transverse field Ising model.