Majorana fermions solve the tetrahedron equations as well as higher simplex equations
Pramod Padmanabhan, Vladimir Korepin
TL;DR
The paper develops a systematic lifting framework to construct higher-$d$-simplex equation solutions from lower-simplex ones, enabling the translation of Yang–Baxter (2-simplex) data into tetrahedron (3-simplex) and higher-dimensional operators. By enforcing a finite set of YM-, YY-, TM-, TY-, TT- and related constraints, the authors show that operators built from Majorana, Dirac, and Clifford-algebra structures satisfy the necessary consistency conditions to lift to arbitrarily high simplices; they also explore anti-Yang–Baxter variants. A key finding is that Clifford-algebra–based solutions can yield positive Boltzmann weights, rendering higher-dimensional integrable statistical-mechanics models physically meaningful, while Majorana constructions offer braid-like actions but do not easily lift within the presented scheme. The work thus provides a versatile toolkit for generating and analyzing higher-simplex integrable structures with potential applications in quantum information and higher-dimensional statistical mechanics.
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.