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 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 quantum inverse scattering method ... a well-developed method ... has allowed a wide class of nonlinear evolution equations to be solved. It explains the algebraic nature of the Bethe ansatz.
The exact solutions of the so-called s-d model (by P.B. Wiegmann in 1980 and independently by N. Andrei, also in 1980) and the Anderson model (by P.B. Wiegmann in 1981, and by N. Kawakami and A. Okiji in 1981) are also both based on the Bethe ansatz. There exist multi-channel generalizations of these two models also amenable to exact solutions (by N. Andrei and C. Destri and by C.J. Bolech and N. Andrei). Recently several models solvable by Bethe ansatz were realized experimentally in solid states and optical lattices. An important role in the theoretical description of these experiments was played by Jean-Sébastien Caux and Alexei Tsvelik.
Example: the Heisenberg antiferromagnetic chain
The Heisenberg antiferromagnetic chain is defined by the Hamiltonian (assuming periodic boundary conditions)
This model is solvable using the Bethe ansatz. The scattering phase shift function is , with in which the momentum has been conveniently reparametrized as in terms of the rapidity. The (here, periodic) boundary conditions impose the Bethe equations
or more conveniently in logarithmic form
where the quantum numbers are distinct half-odd integers for even, integers for odd (with defined mod).
1958: Raymond Lee Orbach uses the Bethe ansatz to solve the Heisenberg model with anisotropic interactions.
1962: J. des Cloizeaux and J. J. Pearson obtain the correct spectrum of the Heisenberg antiferromagnet (spinon dispersion relation), showing that it differs from Anderson's spin-wave theory predictions (the constant prefactor is different).
^Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System". Physical Review. 150 (1): 321-327. Bibcode:1966PhRv..150..321Y. doi:10.1103/PhysRev.150.321.
^Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System". Physical Review. 150 (1): 327-339. Bibcode:1966PhRv..150..327Y. doi:10.1103/PhysRev.150.327.
^Yang, C. N.; Yang, C. P. (July 1969). "Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction". Journal of Mathematical Physics. 10 (7): 1115-1122. Bibcode:1969JMP....10.1115Y. doi:10.1063/1.1664947.