Symmetries and exact solutions of some integrable Haldane-Shastry like spin chains

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 $Y(g{l}_M)$ 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 $Y(g{l}_M)$ Yangian algebra. By using this conjecture, and acting descendent operator on the highest weight states associated with a nonstandard $Y(g{l}_2)$ Yangian algebra, we are able to find out the complete set of eigenvalues and eigenfunctions for the related HS like spin-$\frac{1}{2}$ 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 $Y(g{l}_2)$ Yangian symmetry.

35 pages, latex, no figures, minor type errors are corrected, version to appear in Nucl. Phys. B

References in corpus (2)

• Exactly integrable family of generalized Hubbard models with twisted Yangian symmetry
• Multi-parameter deformed and nonstandard $Y(g{l}_M)$ Yangian symmetry in a novel class of spin Calogero-Sutherland models

Cited by in corpus (16)

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• Exact spectrum and partition function of SU(m|n) supersymmetric Polychronakos model
• Boson-fermion duality in SU(m|n) supersymmetric Haldane-Shastry spin chain
• Thermodynamics of spin chains of Haldane-Shastry type and one-dimensional vertex models
• Spectral properties of supersymmetric Polychronakos spin chain associated with A_{N-1} root system
• The spin Sutherland model of D_N type and its associated spin chain
• Level density distribution for one-dimensional vertex models related to Haldane-Shastry like spin chains
• Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators
• Appearance of branched motifs in the spectra of $B{C}_N$ type Polychronakos spin chains
• Supersymmetric analogue of BC_N type rational integrable models with polarized spin reversal operators
• A novel class of translationally invariant spin chains with long-range interactions
• Super Rogers-Szegö polynomials associated with $B{C}_N$ type of Polychronakos spin chains
• Second Quantized Reduced Bloch Equations and the Exact Solutions for Pairing Hamiltonian