Quantum spin chains with lattice supersymmetry
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This thesis has studied spin chains with Lattice supersymmetry. Connections have been found between Lattice supersymmetry and integrability. In particular, the construction of lattice supercharge has been incorporated into the construction of the quantum inverse scattering method. The open XXZ chain Hamiltonian has been studied, and the construction of the supersymmetric open XXZ chain Hamiltonian by C. Hagendorf and J. Lienardy has been extended. The integrable open XXZ chain Hamiltonian with generic boundary matrices has been studied using TQ equations proposed by W.Yang, R. I. Nepomechie and Y.Zhang, and the integrable open XXZ chain Hamiltonian with diagonal boundary matrices has been studied using the algebraic Bethe ansatz developed by E. K. Sklyanin. For these two types of Hamiltonians, conditions on the boundary parameters have been found where Hamiltonians will have a symmetry in their Bethe equations. This symmetry induces a map SBethe, which shares certain properties with the lattice supercharge SN. An open XXZ chain Hamiltonian has been found which has symmetry with respect to both SN and SBethe. This Hamiltonian is denoted as H(N)XXZ,diagonal. Using the numerical method developed by B.McCoy, the action of the map SBethe of H(N)XXZ,diagonal has been calculated, and the result has conﬁrmed numerically that SBethe is proportional to the lattice supercharge SN. Motivated by this connection, a family of commutation relations between the lattice supercharge of H(N)XXZ,diagonal and sub-matrices of the monodromy matrix corresponding to H(N)XXZ,diagonal have been found and proved using diagrammatic method. Similar commutation relations have also been obtained for the open XXZ chain Hamiltonian with anti-diagonal boundary matrices.