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

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 $U_q(sl_2)$. We analyse the spectra further by using both the coordinate Bethe ansatz and a functional approach, and show that the degeneracies result from special exact $n$-string solutions of the Bethe equations. We also find a family of commuting chiral Hamiltonians that break the degeneracies and allow an integrable interpolation between ferro- and antiferromagnets.