Bethe ansatz

In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. Since then the method has been extended to other spin chains and statistical lattice models. "Bethe ansatz problems" were one of the topics featuring in the "To learn" section of Richard Feynman's blackboard at the time of his death. In the framework of many-body quantum mechanics, models solvable by the Bethe ansatz can be contrasted with free fermion models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions (bosons) is the anti-symmetrized (symmetrized) product of one-body wave functions. Models solvable by the Bethe ansatz are not free: the two-body sector has a non-trivial scattering matrix, which in general depends on the momenta. On the other hand, the dynamics of the models solvable by the Bethe ansatz is two-body reducible: the many-body scattering matrix is a product of two-body scattering matrices. Many-body collisions happen as a sequence of two-body collisions and the many-body wave function can be represented in a form which contains only elements from two-body wave functions. The many-body scattering matrix is equal to the product of pairwise scattering matrices. The generic form of the (coordinate) Bethe ansatz for a many-body wavefunction is in which is the number of particles, their position, is the set of all permutations of the integers , is the (quasi-)momentum of the -th particle, is the scattering phase shift function and is the sign function. This form is universal (at least for non-nested systems), with the momentum and scattering functions being model-dependent. The Yang–Baxter equation guarantees consistency of the construction. The Pauli exclusion principle is valid for models solvable by the Bethe ansatz, even for models of interacting bosons.