In physics, the Bethe ansatz is an ansatz 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, etc.
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 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 ground state is a Fermi sphere. Periodic boundary conditions lead to the Bethe ansatz equations. In logarithmic form the Bethe ansatz equations can be generated by the Yang action. The square of the norm of Bethe wave function is equal to the determinant of the matrix of second derivatives of the Yang action. The recently[when?] developed algebraic Bethe ansatz (see the 1993 book, revised 1997, "Quantum Inverse Scattering Method and Correlation Functions" ) led to essential progress, stating that "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 .
- V. E. Korepin. Calculation of norms of Bethe wave functions. Comm. Math. Phys. 86 3, 391-418 (1982) 
- Quantum Inverse Scattering Method and Correlation Functions
- P.B. Wiegmann, Soviet Phys. JETP Lett., 31, 392 (1980).
- N. Andrei, Phys. Rev. Lett., 45, 379 (1980). APS
- P.B. Wiegmann, Phys. Lett. A 80, 163 (1981). ScienceDirect
- N. Kawakami, and A. Okiji, Phys. Lett. A 86, 483 (1981). ScienceDirect
- N. Andrei and C. Destri Phys. Rev. Lett., 52, 364 (1984). APS
- C.J. Bolech and N. Andrei Phys. Rev. Lett., 88, 237206 (2002). APS
- H. Bethe (1931). "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette". (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain), Zeitschrift für Physik, 71:205–226 (1931). SpringerLink.