Landscapes of integrable long-range spin chains

TL;DR

This work analyzes the elliptic long-range spin chain of Matushko and Zotov (MZ) through a modified, undeformed variant (MZ') and compares its vertex-type landscape to the face-type q-deformed Inozemtsev chain KL_23. It shows that, although the two constructions share structural similarities and possess analogous long-range and macroscopic limits, they are not globally connected by a face-vertex transformation; the SZ' undeformed limit sits at a distinct point and, in the long-range regime, SZ' maps to an antiperiodic Fukui–Kawakami chain. Wrapping and perturbative expansions around nearest-neighbour limits reveal how HS-like physics emerges in the rational limit, while preserving different boundary conditions and symmetry algebras on the vertex vs face sides. The results clarify the relationship between elliptic vertex-type chains and their face-type counterparts, highlighting a single common intersection at the rational HS chain and outlining avenues for exact spectra via elliptic quantum groups and Bethe-ansatz techniques. The paper thus delineates a rich, distinct landscape of long-range integrable spin chains, with clear implications for exact solvability and the role of boundary twists in elliptic settings.

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. The limit $q\to1$ 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.

Figures (3)

• Figure 1: Landscape given by \ref{['eq:V_intro']}, parametrised by periods $(N,\omega)$, for us $\in \mathbb{N}_{\geqslant 2} \times \mathrm{i}\,\mathbb{R}_{>0}$ and $\omega = \mathrm{i} \pi/\kappa$. The limit $\omega \to \mathrm{i} \, 0^+$ requires a normalising factor $n_\kappa$, e.g. $n_\kappa = \sinh^2(\kappa)/\kappa^2$, and that $u \in \mathbb{R}$ with $|u|\geqslant 1$.
• Figure 2: Landscape of the elliptic Matushko--Zotov chain, with the Sechin--Zotov and (antiperiodic) Fukui--Kawakami chains at the undeformed level. In the long-range limits we use a spin rotation. The short-range limit is the xx chain with antiperiodic boundary conditions, deformed for $\eta\neq 0$.
• Figure 3: Landscape of the q-deformed Inozemtsev chain. The undeformed level contains the Heisenberg, Inozemtsev and Haldane--Shastry chains.