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arXiv:hep-th/9110074 · Physics Letters B

A two-parameter deformation of the SU (1|1) superalgebra and the XY quantum chain in a magnetic field

Haye Hinrichsen, Vladimir Rittenberg

Published Jan 1, 1992• 43 citations
arXivHTMLPDF
DOI
10.1016/0370-2693(92)91601-5

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We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the SU(1/1)SU(1/1)SU(1/1) superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.
We examine in this paper the two parameter deformed superalgebra 𝒰qs(sl(1‖2)) and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the R-matrix scheme of Faddeev, Reshetikhin, and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers p≥2 in the undeformed case and by p∈Z in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions of SLqs(1‖2) is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed t−J models.