2005/04/13 by Davide Fioravanti, Marco Rossi
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Complex plane #Eigenvalues and eigenvectors #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum field theory #Quantum mechanics #Scattering #Scattering amplitude #Transfer matrix #cond-mat.stat-mech #hep-th #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1088/1126-6708/2005/08/010
published as JHEP0508:010,2005 · Article, 41 pages, Latex
arxiv created 2005/04/13 · openalex publication_date 2005/08/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the \it disordered regime, thus paralleling similar results by Klümper \citeKLU, achieved through a different technique in the \it antiferroelectric regime. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther \citeLUT and Johnson et al. \citeJKM. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. \citeSKAO. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov \citeZAMe and Lukyanov-Mussardo-Penati \citeLUK, MP as plausible scattering description of unknown integrable field theories.