, the Bethe ansatz
is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe
in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model
Hamiltonian. Since then the method has been extended to other models in one dimension: Bose gas
, Hubbard model
In the framework of many-body quantum mechanics
, models solvable by the Bethe ansatz can be compared to free fermion models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions
) 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 collision 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 Yang-Baxter equation
guarantees the consistency. Experts conjecture that each universality class
in one dimension contains at least one model solvable by the Bethe ansatz. The Pauli exclusion......