Fukui-Kawakami chains: spectrum and hidden $\mathfrak{gl}(1|1)$-symmetry
Fukui and Kawakami showed that the trigonometric Haldane--Shastry (HS) spin chain can be deformed by introducing twisted boundary conditions. In this work we revisit the resulting twisted HS chains, which we call Fukui--Kawakami (FK) chains. We analyse their spectra in dependence of the twist parameter, utilising a direct connection with the (untwisted) HS chain. This allows us to explain why part of the spectrum can be described by Fukui and Kawakami's twisted Bethe equations, and can be constructed from Yangian highest weight states. These states can be labelled by combinatorial data called `motifs', as in the HS chain, which cover some of the (deformed) descendants. We furthermore show that there are other descendants which do not follow from these Bethe equations, but whose energy can be described as a sum of two single-particle energies, suggesting additional hidden symmetries. We then focus on the special case of antiperiodic boundary conditions, and show that this chain coincides with the `minimally polarised' long-range model recently introduced by Basu-Mallick, Finkel, and Gonz\'alez-L\'opez. Remarkably, this connection implies that the antiperiodic chain has a hidden $\mathfrak{gl}(1|1)$-symmetry, which we use to relate the HS motifs to their `supersymmetric' $\mathfrak{gl}(1|1)$ counterparts.