New type of solutions of Yang-Baxter equations, quantum entanglement and related physical models

TL;DR

This paper introduces a new class of Yang-Baxter equation solutions, called type-II, which are linked to quantum entanglement, physical models like the Kitaev Hamiltonian, and differ fundamentally from traditional solutions in their mathematical and physical properties.

Contribution

The paper presents the discovery of type-II solutions to the Yang-Baxter equation, their relation to quantum entanglement, and their application to models like the Kitaev Hamiltonian and parafermion systems, expanding the theoretical framework.

Findings

Type-II solutions differ from type-I in mathematical structure and physical implications.

Maximum entanglement corresponds to the extremum of the introduced $ ext{l}_1$-norm.

Type-II solutions suggest alternative descriptions for quantum information and chain model velocities.

Abstract

Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced $\ell_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to $\mathbb{Z}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $\frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for $\mathbb{Z}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces $\{1,\frac{1}{2},\frac{1}{2}\}$.