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.

Discussion
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



\Psi_M(j_1, \cdots, j_M) = \prod_{M \geq a > b \geq 1} \text{sgn}(j_a - j_b) \sum_{P \in \mathfrak S_M} (-1)^{[P]} \exp \left(i \sum_{a=1}^M k_{P_a} j_a + \frac{i}{2}\sum_{M \geq a > b \geq 1} \mathrm{sgn}(j_a - j_b) \phi(k_{P_a}, k_{P_b})\right) $$

in which $$M$$ is the number of particles, $$j_a, a=1, \cdots M$$ their position, $$\mathfrak S_M$$ is the set of all permutations of the integers $$1, \cdots, M$$, $$(-1) ^{[P]} $$ is the parity of the permutation $$P$$ taking values either positive or negative one, $$k_a$$ is the (quasi-)momentum of the $$a$$-th particle, $$\phi$$ is the scattering phase shift function and $$\mathrm{sgn}$$ 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.

The ground state is a Fermi sphere. Periodic boundary conditions lead to the Bethe ansatz equations or simply Bethe 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 Hessian of the Yang action.

A substantial generalization is the quantum inverse scattering method, or algebraic Bethe ansatz, which gives an ansatz for the underlying operator algebra that "has allowed a wide class of nonlinear evolution equations to be solved."

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.

Terminology
There are many similar methods which come under the name of Bethe ansatz
 * Algebraic Bethe ansatz. The quantum inverse scattering method is the method of solution by algebraic Bethe ansatz, and the two are practically synonymous.
 * Analytic Bethe ansatz
 * Coordinate Bethe ansatz
 * Functional Bethe ansatz
 * Nested Bethe ansatz
 * Thermodynamic Bethe ansatz

Heisenberg antiferromagnetic chain
The Heisenberg antiferromagnetic chain is defined by the Hamiltonian (assuming periodic boundary conditions)



H = J \sum_{j=1}^N \mathbf{S}_{j} \cdot \mathbf{S}_{j+1}, \qquad \mathbf{S}_{j+N} \equiv \mathbf{S}_j. $$

This model is solvable using the (coordinate) Bethe ansatz. The scattering phase shift function is $$\phi(k_a(\lambda_a), k_b(\lambda_b)) = \theta_2 (\lambda_a - \lambda_b)$$, with $$\theta_n (\lambda) \equiv 2 \arctan \frac{2\lambda}{n}$$ in which the momentum has been conveniently reparametrized as $$k(\lambda) = \pi - 2 \arctan 2\lambda$$ in terms of the rapidity $$\lambda.$$ The (here, periodic) boundary conditions impose the Bethe equations



\left[ \frac{ \lambda_a + i/2}{\lambda_a - i/2} \right]^N = \prod_{b \neq a}^M \frac{\lambda_a - \lambda_b + i}{\lambda_a - \lambda_b - i}, \qquad a = 1, ..., M $$

or more conveniently in logarithmic form



\theta_1(\lambda_a) - \frac{1}{N} \sum_{b = 1}^M \theta_2(\lambda_a - \lambda_b) = 2\pi \frac{I_a}{N} $$

where the quantum numbers $$I_j$$ are distinct half-odd integers for $$N - M$$ even, integers for $$N - M$$ odd (with $$I_j$$ defined mod$$(N)$$).

Applicability
The following systems can be solved using the Bethe ansatz
 * Anderson impurity model
 * Gaudin model
 * XXX and XXZ Heisenberg spin chain for arbitrary spin $$s$$
 * Hubbard model
 * Kondo model
 * Lieb–Liniger model
 * Six-vertex model and Eight-vertex model (through Heisenberg spin chain)

Chronology

 * 1928: Werner Heisenberg publishes his model.
 * 1930: Felix Bloch proposes an oversimplified ansatz which miscounts the number of solutions to the Schrödinger equation for the Heisenberg chain.
 * 1931: Hans Bethe proposes the correct ansatz and carefully shows that it yields the correct number of eigenfunctions.
 * 1938: obtains the exact ground-state energy of the Heisenberg model.
 * 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).
 * 1963: Elliott H. Lieb and Werner Liniger provide the exact solution of the 1d δ-function interacting Bose gas (now known as the Lieb-Liniger model). Lieb studies the spectrum and defines two basic types of excitations.
 * 1964: Robert B. Griffiths obtains the magnetization curve of the Heisenberg model at zero temperature.
 * 1966: C.N. Yang and C.P. Yang rigorously prove that the ground-state of the Heisenberg chain is given by the Bethe ansatz. They study properties and applications in and.
 * 1967: C.N. Yang generalizes Lieb and Liniger's solution of the δ-function interacting Bose gas to arbitrary permutation symmetry of the wavefunction, giving birth to the nested Bethe ansatz.
 * 1968: Elliott H. Lieb and F. Y. Wu solve the 1d Hubbard model.
 * 1969: C.N. Yang and C.P. Yang obtain the thermodynamics of the Lieb-Liniger model, providing the basis of the thermodynamic Bethe ansatz (TBA).