Quantum inverse scattering method
Encyclopedia
Quantum inverse scattering method relates two different approaches:
1) Inverse scattering transform
is a method of solving classical integrable differential equations of evolutionary type.
Important concept is Lax representation.
2) Bethe ansatz
is a method of solving quantum models in one space and one time dimension.
Quantum inverse scattering method starts by quantization of Lax representation and reproduce results of Bethe ansatz.
Actually it permits to rewrite Bethe ansatz in a new form: algebraic Bethe ansatz. This led to further progress in understanding of quantum Integrable system
for example a) Heisenberg model (quantum)
,
b) quantum Nonlinear Schrödinger equation
(also known as Lieb-Liniger Model
or Tonks–Girardeau gas) and c) Hubbard model
... Theory of correlation functions was developed: determinant representations, description by differential equations and Riemann-Hilbert problem. Asymptotic of correlation functions (even for space, time and temperature dependent) was evaluated in 1991. Explicit expression for higher conservation laws was obtained in 1989. In mathematics quantum inverse scattering method led to formulation of quantum groups. Especially interesting is Yangian
, the center of the Yangian is given by quantum determinant. Essential progress was achieved in study of Ice-type model
: the bulk free energy of six vertex model depends on boundary conditions even in thermodynamic limit
.
In mathematics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions introduced by L. D. Faddeev in about 1979.
1) Inverse scattering transform
Inverse scattering transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...
is a method of solving classical integrable differential equations of evolutionary type.
Important concept is Lax representation.
2) Bethe ansatz
Bethe ansatz
In physics, the Bethe ansatz is a 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...
is a method of solving quantum models in one space and one time dimension.
Quantum inverse scattering method starts by quantization of Lax representation and reproduce results of Bethe ansatz.
Actually it permits to rewrite Bethe ansatz in a new form: algebraic Bethe ansatz. This led to further progress in understanding of quantum Integrable system
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
for example a) Heisenberg model (quantum)
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...
,
b) quantum Nonlinear Schrödinger equation
Nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
(also known as Lieb-Liniger Model
Lieb-Liniger Model
The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.-Introduction:A model of a gas of particles moving in one dimension and satisfying...
or Tonks–Girardeau gas) and c) 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...
... Theory of correlation functions was developed: determinant representations, description by differential equations and Riemann-Hilbert problem. Asymptotic of correlation functions (even for space, time and temperature dependent) was evaluated in 1991. Explicit expression for higher conservation laws was obtained in 1989. In mathematics quantum inverse scattering method led to formulation of quantum groups. Especially interesting is Yangian
Yangian
Yangian is an important structure in modern representation theory, a type of a quantum group with origins in physics. Yangians first appeared in the work of Ludvig Faddeev and his school concerning the quantum inverse scattering method in the late 1970s and early 1980s. Initially they were...
, the center of the Yangian is given by quantum determinant. Essential progress was achieved in study of Ice-type model
Ice-type model
In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. Variants have been proposed as models of certain...
: the bulk free energy of six vertex model depends on boundary conditions even in thermodynamic limit
Thermodynamic limit
In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...
.
In mathematics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions introduced by L. D. Faddeev in about 1979.