Quantum inverse scattering method: Difference between revisions

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{{Short description|Method used to solve integrable many-body quantum systems}}

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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.{{Citation needed|date=September 2024}}

#the [[inverse scattering transform]], a method of solving classical integrable differential equations of the evolutionary type.{{Citation needed|date=September 2024}}

This method led to the formulation of [[quantum group]]s, in particular the [[Yangian]].{{Citation needed|date=September 2024}} The center of the Yangian, given by the [[quantum determinant]] plays a prominent role in the method.{{Citation needed|date=September 2024}}

An important concept in the [[inverse scattering transform]] is the [[Lax pair|Lax representation]]. The quantum inverse scattering method starts by the [[quantization (physics)|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''.<ref>See for example lectures by N.A. Slavnov {{arXiv|1804.07350}}</ref> This led to further progress in the understanding of quantum [[integrable system]]s, 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]].{{Citation needed|date=September 2024}}

The theory of [[correlationCorrelation function (statistical mechanics)|correlation functions]]s was developed{{when|date=November 2015}}, 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.{{Citation needed|date=September 2024}}

Explicit expressions for the higher [[conservation law]]s of the integrable models were obtained in 1989.{{Citation needed|date=September 2024}}

Essential progress was achieved in study of [[ice-type model]]s: the bulk free energy of the

six vertex model depends on boundary conditions even in the [[thermodynamic limit]].{{Citation needed|date=September 2024}}

==Procedure==