Bethe ansatz


In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly in one spatial dimension. It was introduced by Hans Bethe in 1931 to obtain the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic Heisenberg model.
The approach was later generalized into the quantum inverse scattering method and the algebraic Bethe ansatz, forming the basis of modern integrable system theory.
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. The dynamics of a free model is one-body reducible: its many-body wave function for fermions is the antisymmetrized product of one-body wave functions. Models solvable by the Bethe ansatz are interacting systems: their two-body sector has a nontrivial scattering matrix that depends on the particle momenta.
By contrast, such models are two-body reducible: the many-body scattering matrix factorizes into a product of two-body scattering matrices. Many-body collisions occur as sequences of pairwise interactions, and the total wave function can be represented entirely in terms of two-body scattering states. The overall scattering matrix equals the ordered product of these pairwise matrices.
where is the number of particles, are their position, is the set of all permutations of the integers ; is the parity of the permutation ; is the momentum of the -th particle, is the scattering phase shift function and is the sign function. This form is universal, 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 and the Anderson model are also both based on the Bethe ansatz. There exist multi-channel generalizations of these two models also amenable to exact solutions. 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

Examples

Heisenberg antiferromagnetic chain

The Heisenberg antiferromagnetic chain is defined by the Hamiltonian
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 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.

Applicability

The following systems can be solved using the Bethe ansatz

Chronology