Symbolic evaluation of transfer matrices for the XXX model

We aim to exploit the important fact that a spin chain has nowadays the status of a quantum integrable system: essentially, its transfer matrix involves a full information on the constants of motion which provides an exact determination of all eigenstates of this system. Here we consider the specific case of the XXX 1/2 model, whose spherical symmetry, and thus a relative numerical simplicity, admits a straightforward presentation of results. This case is also specific as a quantum integrable system since the eigenproblem of its transfer matrix exhibits multiplets of the total spin, where each highest weight “ancestor” state is degenerate with all its “descendants”, and thus the z-component of the total spin should be added to integrability constants of motion to form a complete system of commuting observables. We provide here an efficient algorithm for evaluation of all elements, A(λ), B(λ), C(λ), and D(λ), of the monodromy matrix M(λ) of the XXX 1/2 model, defined within the famous algebraic Bethe Ansatz (ABA) formalism. Accordingly, the transfer matrix T(λ) will be explicitly presented as the N-th degree polynomial in the rapidity λ, with coefficients T(λ) being Hermitian operators (constants of motion), whose spectra determine each eigenstate of the system. In this way, all eigenstates of the model can be determined by an immediate diagonalization of the transfer matrix, that is, by N linear eigenproblems, instead of solving a cumbersome system of r nonlinear Bethe Ansatz equations, with N being the size of a chain, and r - the number of inverted spins in an eigenstate.We believe that exact results, reached in this way for moderate sizes N, will be helpful in quantum information processing and nanotechnology. The algorithm bases on an observation that the monodromy matrix is the sum of some elementary operators, referred by us as to “gear racks”, with, essentially, single-particle nature. We provide here a complete combinatoric description of these objects, and appropriate programming for determination of the transfer matrix T(λ), some its important submatrices, as well as creation B(λ) and annihilation C(λ) operators of Bethe pseudoparticles.