The report DESY‑24‑062, jointly produced by the physics and mathematics groups at DESY and Humboldt University, investigates the recently discovered elliptic integrable spin chain of Matushko and Zotov (MZ). The authors, Rob Klabbers and Jules Lamers, analyse how this chain fits into the broader family of long‑range integrable models and determine its various limiting behaviours.

The MZ chain is defined on a periodic lattice of (N) spin‑½ sites and is built from an elliptic (R)‑matrix of Baxter’s eight‑vertex type. By introducing a deformation parameter (q) the authors construct a modified chain, denoted (MZ’), whose Hamiltonian contains chiral nearest‑neighbour terms and a non‑trivial deformed translation operator. In the short‑range limit (qto 0) the model reduces to the standard XX chain with (q)‑deformed antiperiodic boundary conditions. Taking the further limit (qto 1) yields the elliptic spin chain of Sechin and Zotov (SZ). The SZ chain, in its trigonometric specialization, coincides with the model of Fukui and Kawakami. Both SZ and its trigonometric limit admit a short‑range limit that is again the antiperiodic XX model.

A key technical result is the explicit identification of the translation operator for the (MZ’) chain. Its non‑trivial structure guarantees that antiperiodicity is a persistent feature of the entire vertex‑type landscape. The authors compare this landscape with the face‑type landscape that contains the Heisenberg XXX and the Haldane–Shastry chains. They find that the two landscapes intersect only at the rational Haldane–Shastry point, indicating that the elliptic MZ chain is genuinely distinct from the face‑type models.

Using a wrapping argument the report shows that the SZ chain is an antiperiodic version of the Inozemtsev chain. The Inozemtsev Hamiltonian, which involves the Weierstrass elliptic function (V(u)=wp(u)+text{const}), interpolates between nearest‑neighbour Heisenberg interactions and long‑range (1/sin^2) or (1/sinh^2) potentials depending on the periods. By expanding both the SZ and Inozemtsev chains around their nearest‑neighbour limits the authors provide a clear interpretation of these models as long‑range deformations of the Heisenberg chain.

The technical analysis relies on a careful study of elliptic functions and their limits, the properties of Baxter’s eight‑vertex (R)‑matrix, and the face‑vertex transformation. The authors derive explicit expressions for the trigonometric and short‑range limits of the (R)‑matrix, and show how the deformation parameter (q) controls the boundary conditions. The work also clarifies the role of the twist in the translation operator and its effect on the spectrum.

Collaboration-wise, the project is a joint effort between DESY’s physics department and Humboldt University’s mathematics department, with contributions from both the physics and mathematics groups. The report was produced in 2024 and is part of a broader research programme on integrable systems and long‑range spin chains. The authors acknowledge support from the German research funding agencies that sponsor interdisciplinary projects between physics and mathematics.