dorsal/arxiv
View SchemaSymmetries and exact solutions of some integrable Haldane-Shastry like spin chains
| Authors | B. Basu-Mallick |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9809010 |
| URL | https://arxiv.org/abs/solv-int/9809010 |
| DOI | 10.1016/S0550-3213(98)00784-6 |
Abstract
By using a class of `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry (HS) spin chain. Lax pairs and conserved quantities for these spin chains are also found and it is established that these models exhibit multi-parameter deformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by projecting the eigenstates of Dunkl operators in a suitable way, we derive a class of exact eigenfunctions for such HS like spin chain and subsequently conjecture that these exact eigenfunctions would lead to the highest weight states associated with a multi-parameter deformed or nonstandard variant of $Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent operator on the highest weight states associated with a nonstandard $Y(gl_2)$ Yangian algebra, we are able to find out the complete set of eigenvalues and eigenfunctions for the related HS like spin-${1\over 2}$ chain. It turns out that some additional energy levels, which are forbidden due to a selection rule in the case of SU(2) HS model, interestingly appear in the spectrum of above mentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.
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"abstract": "By using a class of `anyon like\u0027 representations of permutation algebra,\nwhich pick up nontrivial phase factors while interchanging the spins of two\nlattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry\n(HS) spin chain. Lax pairs and conserved quantities for these spin chains are\nalso found and it is established that these models exhibit multi-parameter\ndeformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by\nprojecting the eigenstates of Dunkl operators in a suitable way, we derive a\nclass of exact eigenfunctions for such HS like spin chain and subsequently\nconjecture that these exact eigenfunctions would lead to the highest weight\nstates associated with a multi-parameter deformed or nonstandard variant of\n$Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent\noperator on the highest weight states associated with a nonstandard $Y(gl_2)$\nYangian algebra, we are able to find out the complete set of eigenvalues and\neigenfunctions for the related HS like spin-${1\\over 2}$ chain. It turns out\nthat some additional energy levels, which are forbidden due to a selection rule\nin the case of SU(2) HS model, interestingly appear in the spectrum of above\nmentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.",
"arxiv_id": "solv-int/9809010",
"authors": [
"B. Basu-Mallick"
],
"categories": [
"solv-int",
"cond-mat.stat-mech",
"hep-th",
"nlin.SI"
],
"doi": "10.1016/S0550-3213(98)00784-6",
"title": "Symmetries and exact solutions of some integrable Haldane-Shastry like spin chains",
"url": "https://arxiv.org/abs/solv-int/9809010"
},
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