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In quantum physics, the quantum inverse scattering method (QISM) or the algebraic Bethe ansatz is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979. [1]
It can be viewed as a quantized version of the classical inverse scattering method pioneered by Norman Zabusky and Martin Kruskal [2] used to investigate the Korteweg–de Vries equation and later other integrable partial differential equations. In both, a Lax matrix features heavily and scattering data is used to construct solutions to the original system.
While the classical inverse scattering method is used to solve integrable partial differential equations which model continuous media (for example, the KdV equation models shallow water waves), the QISM is used to solve many-body quantum systems, sometimes known as spin chains, of which the Heisenberg spin chain is the best-studied and most famous example. These are typically discrete systems, with particles fixed at different points of a lattice, but limits of results obtained by the QISM can give predictions even for field theories defined on a continuum, such as the quantum sine-Gordon model.
The quantum inverse scattering method relates two different approaches:
This method led to the formulation of quantum groups, in particular the Yangian. The center of the Yangian, given by the quantum determinant plays a prominent role in the method.
An important concept in the inverse scattering transform is the Lax representation. 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. [3] This led to further progress in the understanding of quantum integrable systems, such as the quantum Heisenberg model, the quantum nonlinear Schrödinger equation (also known as the Lieb–Liniger model or the Tonks–Girardeau gas) and the Hubbard model.
The theory of correlation functions was developed[ when? ], relating determinant representations, descriptions by differential equations and the Riemann–Hilbert problem. Asymptotics of correlation functions which include space, time and temperature dependence were evaluated in 1991.
Explicit expressions for the higher conservation laws of the integrable models were obtained in 1989.
Essential progress was achieved in study of ice-type models: the bulk free energy of the six vertex model depends on boundary conditions even in the thermodynamic limit.
The steps can be summarized as follows (EvgenySklyanin 1992 ):
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation for a function dependent on two variables typically denoted and , involving the wave operator and the sine of .
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation.
Martin David Kruskal was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His most celebrated contribution was in the theory of solitons.
In physics and mathematics, an ansatz is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.
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 nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an 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 integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.
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.
Ludvig Dmitrievich Faddeev was a Soviet and Russian mathematical physicist. He is known for the discovery of the Faddeev equations in the quantum-mechanical three-body problem and for the development of path-integral methods in the quantization of non-abelian gauge field theories, including the introduction of the Faddeev–Popov ghosts. He led the Leningrad School, in which he along with many of his students developed the quantum inverse scattering method for studying quantum integrable systems in one space and one time dimension. This work led to the invention of quantum groups by Drinfeld and Jimbo.
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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 mechanically. It is related to the prototypical Ising model, where at each site of a lattice, a spin represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction.
In mathematics, the inverse scattering transform is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical methods related to wave scattering. The direct scattering transform describes how a function scatters waves or generates bound-states. The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering. The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations.
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 solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.
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 scattering method. The name Yangian was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang.
Vladimir E. Korepin is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics.
In mathematics, the Novikov–Veselov equation is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Novikov & Veselov (1984).
Leon Armenovich Takhtajan is a Russian mathematical physicist of Armenian descent, currently a professor of mathematics at the Stony Brook University, Stony Brook, NY, and a leading researcher at the Euler International Mathematical Institute, Saint Petersburg, Russia.
In physics, the Gaudin model, sometimes known as the quantum Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also examples of quantum spin chains.
A spin chain is a type of model in statistical physics. Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice. A prototypical example is the quantum Heisenberg model. Interactions between the sites are modelled by operators which act on two different sites, often neighboring sites.
In statistical physics, the Inozemtsev model is a spin chain model, defined on a one-dimensional, periodic lattice. Unlike the prototypical Heisenberg spin chain, which only includes interactions between neighboring sites of the lattice, the Inozemtsev model has long-range interactions, that is, interactions between any pair of sites, regardless of the distance between them.