Baxter's Q-operator for the open XXZ Heisenberg chain with diagonal boundaries

Bart Vlaar (Heriot-Watt University)

Tuesday 5th February, 2019 16:00-17:00 Maths 311B

Abstract

Baxter developed a method for finding the spectrum of the Hamiltonian of the closed XYZ Heisenberg chain, which cannot be done by means of usual algebraic Bethe ansatz techniques. It involved the construction of a family of operators Q(z) commuting with the transfer matrices and, crucially, satisfying a functional equation. Both families of operators act on an N-fold tensor product of vector spaces, in our case finite-dimensional modules over quantum affine sl2, and both are partial traces over an auxiliary vector space. For the transfer matrices this is again a finite-dimensional module over quantum affine sl2. Certain infinite-dimensional modules over its upper and lower Borel subalgebras are used instead for Q(z). The corresponding Q-operator for the open XXZ chain with diagonal boundaries can be obtained using Sklyanin's two-row construction. The key tools for proving the functional equation are short exact sequences of modules over the Borel subalgebras and an algebra automorphism of quantum affine sl2 interchanging the Borel subalgebras in a particular way. By taking scaling limits one expects to recover the Q-operators for the open XXX chain found recently by Frassek and Szécsény. Work in progress, joint with Robert Weston and Alexander Cooper.

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