quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

, the quantum inverse scattering method is a method for solving

integrable model In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

s in 1+1 dimensions, introduced by L. D. Faddeev in 1979. The quantum inverse scattering method relates two different approaches: #the Bethe ansatz, a method of solving integrable quantum models in one space and one time dimension; #the

Inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...

, a method of solving classical integrable differential equations of the evolutionary type. This method led to the formulation of

quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...

s. Especially interesting is the

Yangian In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse ...

, and the center of the Yangian is given by the quantum determinant. An important concept in the

is the

Lax representation In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...

; the quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the ''algebraic Bethe ansatz''.cf. e.g. the lectures by N.A. Slavnov, This led to further progress in the understanding of quantum

Integrable system In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

s, for example: a) the

Heisenberg model (quantum) The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mec ...

, b) the quantum

Nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...

(also known as the

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 Bose–Einstein statistics was introduced in ...

or the

Tonks–Girardeau gas In physics, a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the system's physics. It is named after physicists Marvin D. Girardeau and Lewi Tonks. It is not a B ...

) and c) the

Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each el ...

. The theory of

correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...

s was developed : determinant representations, descriptions by differential equations and the

Riemann–Hilbert problem In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems h ...

. Asymptotics of correlation functions (even for space, time and temperature dependence) were evaluated in 1991. Explicit expressions for the higher

conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...

s of the integrable models were obtained in 1989. Essential progress was achieved in study of

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. ...

s: the bulk free energy of the six vertex model depends on boundary conditions even in the

thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...

References

* *{{Citation , last1=Korepin , first1=V. E. , last2=Bogoliubov , first2=N. M. , last3=Izergin , first3=A. G. , title=Quantum inverse scattering method and correlation functions , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, series=Cambridge Monographs on Mathematical Physics , isbn=978-0-521-37320-3 , mr =1245942 , year=1993 Exactly solvable models Quantum mechanics