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Most of the exact solutions of quantum one-dimensional Hamiltonians areobtained thanks to the success of the Bethe ansatz on its several formulations.According to this ansatz the amplitudes of the eigenfunctions of theHamiltonian are given by a sum of permutations of appropriate plane waves. Inthis paper, alternatively, we present a matrix product ansatz that asserts thatthose amplitudes are given in terms of a matrix product. The eigenvalueequation for the Hamiltonian define the algebraic properties of the matricesdefining the amplitudes. The existence of a consistent algebra imply the exactintegrability of the model. The matrix product ansatz we propose allow anunified and simple formulation of several exact integrable Hamiltonians. Inorder to introduce and illustrate this ansatz we present the exact solutions ofseveral quantum chains with one and two global conservation laws and periodicboundaries such as the XXZ chain, spin-1 Fateev-Zamolodchikov model,Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc.Formulation of the matrix product ansatz for quantum chains with open ends isalso possible. As an illustration we present the exact solution of an extendedXXZ chain with $z$-magnetic fields at the surface and arbitrary hard-coreexclusion among the spins.Comment: 57 pages, no figure

Exact solutions of exactly integrable quantum chains by a matrix product ansatz