Majorana fermions solve the tetrahedron equations as well as higher simplex equations

TL;DR

This paper introduces a systematic lifting method to construct solutions of higher-dimensional simplex equations from lower-dimensional ones, utilizing Majorana fermions and Clifford algebras, with implications for quantum integrable models and statistical mechanics.

Contribution

The authors develop a novel lifting technique to generate solutions of higher simplex equations from lower ones, including solutions based on Majorana and Dirac fermions, advancing the understanding of multi-dimensional integrable models.

Findings

Majorana fermion solutions satisfy higher simplex constraints

Clifford algebra solutions lead to positive Boltzmann weights

Anti-Yang-Baxter operators can be lifted to higher simplex solutions

Abstract

Yang-Baxter equations define quantum integrable models. The tetrahedron and higher simplex equations are multi-dimensional generalizations. Finding the solutions of these equations is a formidable task. In this work we develop a systematic method - constructing higher simplex operators [solutions of corresponding simplex equations] from lower simplex ones. We call it lifting. By starting from solutions of Yang-Baxter equations we can construct solutions of the tetrahedron equation and simplex equation in any dimension. We then generalize this by starting from a solution of any lower simplex equation and lifting it [construct solution] to another simplex equation in higher dimension. This process introduces several constraints among the different lower simplex operators that are lifted to form the higher simplex operators. We show that braided Yang-Baxter operators [solutions of Yang-Baxter equations independent of spectral parameters] constructed using Majorana fermions satisfy these constraints, thus solving the higher simplex equations. As a consequence these solutions help us understand the action of an higher simplex operator on Majorana fermions. Apart from these we show that solutions constructed using Dirac (complex) fermions and Clifford algebras also satisfy these constraints. Furthermore it is observed that the Clifford solutions give rise to positive Boltzmann weights resulting in the possibility of physical statistical mechanics models in higher dimensions. Finally we also show that anti-Yang-Baxter operators [solutions of Yang-Baxter-like equations with a negative sign on the right hand side] can also be lifted to higher simplex solutions.