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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In [[quantum physics]], the '''quantum inverse scattering method''' (QISM) or the '''algebraic Bethe ansatz''' is a method for solving [[integrable system|integrable model]]s in 1+1 dimensions, introduced by [[Leon Takhtajan]] and [[Ludvig Faddeev|L. D. Faddeev]] in 1979.<ref>{{cite journal |last1=Takhtadzhan |first1=L A |last2=Faddeev |first2=Lyudvig D |title=The Quantum Method of the Inverse Problem and the Heisenberg Xyz Model |journal=Russian Mathematical Surveys |date=31 October 1979 |volume=34 |issue=5 |pages=11–68 |doi=10.1070/RM1979v034n05ABEH003909|bibcode=1979RuMaS..34...11T }}</ref>

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In [[quantum physics]], the '''quantum inverse scattering method''' (QISM), orsimilar to the closely related '''algebraic Bethe ansatz''', is a method for solving [[integrable system|integrable model]]s in 1+1 dimensions, introduced by [[Leon Takhtajan]] and [[Ludvig Faddeev|L. D. Faddeev]] in 1979.<ref>{{cite journal |last1=Takhtadzhan |first1=L A |last2=Faddeev |first2=Lyudvig D |title=The Quantum Method of the Inverse Problem and the Heisenberg Xyz Model |journal=Russian Mathematical Surveys |date=31 October 1979 |volume=34 |issue=5 |pages=11–68 |doi=10.1070/RM1979v034n05ABEH003909|bibcode=1979RuMaS..34...11T }}</ref>

It can be viewed as a quantized version of the classical [[inverse scattering method]] pioneered by [[Norman Zabusky]] and [[Martin Kruskal]]<ref>{{cite journal |last1=Zabusky |first1=N. J. |last2=Kruskal |first2=M. D. |title=Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States |journal=Physical Review Letters |date=9 August 1965 |volume=15 |issue=6 |pages=240–243 |doi=10.1103/PhysRevLett.15.240|doi-access=free |bibcode=1965PhRvL..15..240Z }}</ref> used to investigate the [[Korteweg–de Vries equation]] and later other [[integrable system|integrable]] [[partial differential equations]]. In both, a [[Lax matrix]] features heavily and [[scattering|scattering data]] is used to construct solutions to the original system.

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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==