Preprint 2024-01
Rob Klabbers, Jules Lamers
Landscapes of Integrable Long-Range Spin Chains
Abstract:
We clarify how the elliptic integrable spin chain recently found by Matushko and Zotov (MZ) relates to various other known long-range spin chains.
We evaluate various limits. More precisely, we tweak the MZ chain to allow for a short-range limit, and show it is the xx model with q-deformed antiperiodic boundary conditions. Taking q → 1 gives the elliptic spin chain of Sechin and Zotov (SZ), whose trigonometric case is due to Fukui and Kawakami. It, too, can be adjusted to admit a short-range limit, which we demonstrate to be the antiperiodic xx model. By identifying the translation operator of the MZ ′ chain, which is nontrivial, we show that antiperiodicity is a persistent feature.
We compare the resulting (vertex-type) landscape of the MZ chain with the (face-type) landscape containing the Heisenberg xxx and Haldane–Shastry chains. We find that the landscapes only share a single point: the rational Haldane–Shastry chain. Using wrapping we show that the SZ chain is the antiperiodic 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.
37 pp.
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2024-01