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How Coordinate Bethe Ansatz Works for Inozemtsev Model
- Source :
- Communications in Mathematical Physics. 390:827-905
- Publication Year :
- 2022
- Publisher :
- Springer Science and Business Media LLC, 2022.
-
Abstract
- Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest.<br />63 pages, 13 figures, 3 tables. v2: 65 pages, 13 figures, 4 tables. Minor changes. v3: 67 pages, 13 figures, 4 tables. Various minor changes; added discussion of bound states at large $L$, derivation of Haldane's Bethe-ansatz-like equations and discussion of Heisenberg limit of rationalisation
- Subjects :
- High Energy Physics - Theory
Condensed Matter - Strongly Correlated Electrons
Nonlinear Sciences::Exactly Solvable and Integrable Systems
Strongly Correlated Electrons (cond-mat.str-el)
High Energy Physics - Theory (hep-th)
Nonlinear Sciences - Exactly Solvable and Integrable Systems
FOS: Physical sciences
Condensed Matter::Strongly Correlated Electrons
Statistical and Nonlinear Physics
Mathematical Physics (math-ph)
Exactly Solvable and Integrable Systems (nlin.SI)
Mathematical Physics
Subjects
Details
- ISSN :
- 14320916 and 00103616
- Volume :
- 390
- Database :
- OpenAIRE
- Journal :
- Communications in Mathematical Physics
- Accession number :
- edsair.doi.dedup.....6afa492738e4ddfcfa227076b16bef3e