On the supersymmetric XXX spin chains associated to ${\mathfrak{gl}}_{1|1}$

Abstract

We study the ${\mathfrak{gl}}_{1|1}$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation ${\mathfrak{gl}}_{1|1}$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.

arXiv comment32 pages, 1 figure

References in corpus (12)

• Gauss Decomposition of the Yangian Y(gl_{m|n})
• Yangian of the Queer Lie Superalgebra
• Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra
• Representations of Super Yangian
• Nested Bethe ansatz for `all' open spin chains with diagonal boundary conditions
• Multiple Actions of the Monodromy Matrix in $\mathfrak{gl}(2|1)$-Invariant Integrable Models
• Norm of Bethe vectors in models with $\mathfrak{gl}(m|n)$ symmetry
• RTT realization of quantum affine superalgebras and tensor products
• Supersymmetric quantum spin chains and classical integrable systems
• Solutions of ${\mathfrak{gl}}_{m|n}$ XXX Bethe ansatz equation and rational difference operators
• The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}(m|n)$
• The solutions of ${\mathfrak{gl}}_{M|N}$ Bethe ansatz equation and rational pseudodifferential operators

Cited by in corpus (7)

• Jacobi-Trudi identity and Drinfeld functor for super Yangian
• A note on odd reflections of super Yangian and Bethe ansatz
• Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces
• Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra
• Recurrence relations for off-shell Bethe vectors in trigonometric integrable models
• Completeness of Bethe ansatz for Gaudin models associated with gl(1|1)
• On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains