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The Coordinate Bethe Ansatz (CBA) expresses, as a sum over permutations, thematrix element of an XXX Heisenberg spin chain Hamiltonian eigenstate with astate with fixed spins. These matrix elements comprise the wave functions ofthe Hamiltonian eigenstates. However, as the complexity of the sum growsrapidly with the length N of the spin chain, the exact wave function in thecontinuum limit is too cumbersome to be exploited. In this note we provide anapproximation to the CBA whose complexity does not directly depend upon N. Thisconsists of two steps. First, we add an anchor to the argument of theexponential in the CBA. The anchor is a permutation-dependent integral multipleof 2 pi. Once anchored, the distribution of these arguments simplifies,becoming approximately Gaussian. The wave function is given by the Fouriertransform of this distribution and so the calculation of the wave functionreduces to the calculation of the moments of the distribution. Second, weparametrize the permutation group as a map between integers and we bin thesemaps. The calculation of the moments then reduces to a combinatorial exerciseon the partitioning into bins. As an example we consider the matrix elementbetween the classical and quantum ground states.

Anchoring and Binning the Coordinate Bethe Ansatz