Bethe ansatz
Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the Bethe ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...

is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

 in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model
Heisenberg model (quantum)
The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...

 Hamiltonian. Since then the method has been extended to other models in one dimension: Bose gas
Bose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...

, Hubbard model
Hubbard model
The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...

, etc.

In the framework of many-body quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, 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 (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 Yang-Baxter equation
Yang-Baxter equation
The Yang–Baxter equation is an equation which was first introduced in the field of statistical mechanics. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971...

 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 principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

 is valid for models solvable by the Bethe ansatz, even for models of interacting bosons.

The ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

 is a Fermi sphere
Fermi level
The Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...

. 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. Recently developed algebraic Bethe ansatz (see the book ) led to essential progress.

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 an A. Okiji in 1981) are also both based on the Bethe ansatz. Recently several models solvable by Bethe Ansatz we realized experimentally in solid stats and optical lattices, important role in theoretical description of these experiments was played by Jean-Sébastien Caux http://www.science.uva.nl/english/home.cfm/916D5C0A-CFAB-4670-8F18FDC3DD99A3C6 and
Alexei Tsvelik http://www.cmth.bnl.gov/tsvelik.shtml.

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