Onsager symmetries in $U(1)$-invariant clock models

Understand

We show how the Onsager algebra, used in the original solution of the two-dimensional Ising model, arises as an infinite-dimensional symmetry of certain self-dual models that also have a $U(1)$ symmetry.

• We describe in detail the example of nearest-neighbour $n$-state clock chains whose ${\mathbb Z}_n$ symmetry is enhanced to $U(1)$.
• As a consequence of the Onsager-algebra symmetry, the spectrum of these models possesses degeneracies with multiplicities $2^N$ for positive integer $N$.
• We construct the elements of the algebra explicitly from transfer matrices built from non-fundamental representations of the quantum-group algebra $U_q(sl_2)$.