dorsal/arxiv
View Schemaq-Symmetries in DNLS-AL chains and exact solutions of quantum dimers
| Authors | Demosthenes Ellinas, Panagiotis Maniadis |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9907014 |
| URL | https://arxiv.org/abs/quant-ph/9907014 |
| DOI | 10.1142/S0217979299002873 |
| Journal | Int.J.Mod.Phys. B13 (1999) 3087-3106 |
Abstract
Dynamical symmetries of Hamiltonians quantized models of discrete non-linear Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is shown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is given by the $su(n)$ algebra, while the respective symmetry for the AL case is the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of su(n) and su_q(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the $n=2$ quantum dimer case and formulate the eigenvalue problem of each dimer as a non-linear (q)-spin model. Analytic investigations of the ensuing three-term non-linear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of non-linearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the non-linearity parameter near the classical bifurcation point.
{
"annotation_id": "27b2bc92-2e56-45ab-ab4b-3e3772729e25",
"date_created": "2026-03-02T18:02:48.517000Z",
"date_modified": "2026-03-02T18:02:48.517000Z",
"file_hash": "12db596fbc62e0b84f6051f3295f071176454c94851c75c91a95e4bdba44dc75",
"private": false,
"record": {
"abstract": "Dynamical symmetries of Hamiltonians quantized models of discrete non-linear\nSchroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is\nshown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is\ngiven by the $su(n)$ algebra, while the respective symmetry for the AL case is\nthe quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the\nAL model is due to the non-canonical oscillator-like structure of the raising\nand lowering operators at each site.\n Invariants of motions are found in terms of Casimir central elements of su(n)\nand su_q(n) algebra generators, for the DNLS and QAL cases respectively.\nUtilizing the representation theory of the symmetry algebras we specialize to\nthe $n=2$ quantum dimer case and formulate the eigenvalue problem of each dimer\nas a non-linear (q)-spin model. Analytic investigations of the ensuing\nthree-term non-linear recurrence relations are carried out and the respective\northonormal and complete eigenvector bases are determined.\n The quantum manifestation of the classical self-trapping in the QDNLS-dimer\nand its absence in the QAL-dimer, is analysed by studying the asymptotic\nattraction and repulsion respectively, of the energy levels versus the strength\nof non-linearity. Our treatment predicts for the QDNLS-dimer, a\nphase-transition like behaviour in the rate of change of the logarithm of\neigenenergy differences, for values of the non-linearity parameter near the\nclassical bifurcation point.",
"arxiv_id": "quant-ph/9907014",
"authors": [
"Demosthenes Ellinas",
"Panagiotis Maniadis"
],
"categories": [
"quant-ph",
"cond-mat",
"nlin.SI"
],
"doi": "10.1142/S0217979299002873",
"journal_ref": "Int.J.Mod.Phys. B13 (1999) 3087-3106",
"title": "q-Symmetries in DNLS-AL chains and exact solutions of quantum dimers",
"url": "https://arxiv.org/abs/quant-ph/9907014"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "4286d932-bae6-4b92-b73b-3e36d2f6a522",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}