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), similar 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.

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 problem|many-body]] quantum systems, sometimes known as [[spin chain]]s, 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 [[quantum field theory|field theories]] defined on a continuum, such as the quantum [[sine-Gordon model]].

== Discussion ==

The quantum inverse scattering method relates two different approaches:

1) The#the [[Bethe ansatz ]], a method of solving integrable quantum models in one space and one time dimension;.{{Citation needed|date=September 2024}}

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

An important concept in the [[Inverse scattering transform]] is the [[Lax representation]]; the quantum inverse scattering method starts by the quantization of 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. This led to further progress in the understanding of quantum [[Integrable systems]] for example: a) the [[Heisenberg model (quantum)]],

b) the quantum [[Nonlinear Schrödinger equation ]] (also known as the [[Lieb-Liniger Model]] or the [[Tonks–Girardeau gas]]) and c) the [[Hubbard model]]. The

theory of correlation functions was developed: determinant representations, descriptions by differential equations and the [[Riemann-Hilbert problem]].

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

of the integrable models were obtained in 1989. In mathematics the quantum inverse scattering method led to the formulation of [[quantum groups]]. Especially interesting is

the [[Yangian]] and the center of the Yangian is given by the quantum determinant. Essential progress was achieved in study of [[Ice-type_model|ice-type models]]: the bulk free energy of the

six vertex model depends on boundary conditions even in the [[thermodynamic limit]].

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}}

In mathematics, the '''quantum inverse scattering method''' is a method for solving [[integrable model]]s in 1+1 dimensions introduced by [[L. D. Faddeev]] in about 1979.

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 [[Correlation function (statistical mechanics)|correlation functions]] 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}}

the [[Yangian]] and the center of the Yangian is given by the quantum determinant. Essential progress was achieved in study of [[Ice-type_model|ice-type modelsmodel]]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==

The steps can be summarized as follows {{harvs|last=Sklyanin|first=Evgeny|year=1992}}:

# Take an [[R-matrix|''R''-matrix]] which solves the [[Yang–Baxter equation]].

# Take a [[representation (group theory)|representation]] of an algebra <math>\mathcal{T}_R</math> satisfying the RTT{{What|date=October 2023}} relations.{{sfn|Chakrabarti|2001}}

# Find the spectrum of the [[generating function]] <math>t(u)</math> of the [[center (group theory)|centre]] of <math>\mathcal{T}_R</math>.

# Find correlators.

==References==

{{Reflist}}

*{{Citation | last1=Faddeev | first1=L. | title=Instructive history of the quantum inverse scattering method |doi=10.1007/BF00994626 | mr = 1329554 | year=1995 | journal=Acta Applicandae Mathematicae | volume=39 | issue=1 | pages=69–84}}

*{{cite journal |last1=Chakrabarti |first1=A. |title=RTT relations, a modified braid equation and noncommutative planes |journal=Journal of Mathematical Physics |year=2001 |volume=42 |issue=6 |pages=2653–2666 |doi=10.1063/1.1365952 |url=https://pubs.aip.org/aip/jmp/article-abstract/42/6/2653/803265/RTT-relations-a-modified-braid-equation-and|arxiv=math/0009178 |bibcode=2001JMP....42.2653C }}

*{{cite arXiv |eprint=hep-th/9211111|last1=Sklyanin|first1=E. K.|title=Quantum Inverse Scattering Method. Selected Topics|year=1992}}

*{{Citation | last1=Faddeev | first1=L. | title=Instructive history of the quantum inverse scattering method |doi=10.1007/BF00994626 | mr = 1329554 | year=1995 | journal=Acta Applicandae Mathematicae | volume=39 | issue=1 | pages=69–84| s2cid=120648929 }}

*{{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]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-37320-3 | mr =1245942 | year=1993}}

{{Integrable systems}}

[[Category:Exactly solvable models]]

[[Category:Quantum mechanics]]