Landscapes of integrable long-range spin chains

We clarify how the elliptic integrable spin chain recently found by Matushko and Zotov (MZ) relates to various other known long-range spin chains. The limit $q\to 1$ gives the elliptic spin chain of Sechin and Zotov (SZ), whose trigonometric case is due to Fukui and Kawakami. At finite size, only the latter is U(1)-symmetric. We compare the resulting (vertex-type) landscape of the MZ chain with the (face-type) landscape containing the Heisenberg XXX and Haldane–Shastry (HS) chains, as well as the Inozemtsev chain and its recent q-deformation. We find that the two landscapes only share a single point: the rational HS chain. Using wrapping we show that the SZ chain is the anti-periodic version of the Inozemtsev chain in a precise sense, and expand both chains around their nearest-neighbour limits to facilitate their interpretations as long-range deformations.