Solvable Models with Self-Triality in Statistical Mechanics and Field Theory

The notion of self-duality is extended to self-triality. One example from spin systems is given and completely solved by use of fermion variables. It is then shown that the O(8) Gross-Neveu model has self-triality: The Lagrangian $\mathcal{L}\left(\psi \right)=\mathcal{L}\mathrm{}\left(R\right)=\mathcal{L}\mathrm{}\left(L\right)\mathrm{}$, where $\psi$ is the original fermion while $R$ and $L$ are two types of kinks that occur dynamically. The anatomy of self-duality (triality) in the Ising and present examples is exposed as is the origin of the fermionic solutions.