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}}</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=THEThe QUANTUMQuantum METHODMethod OFof THEthe INVERSEInverse PROBLEMProblem ANDand THEthe HEISENBERGHeisenberg XYZXyz MODELModel |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]].

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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 [[Nonlinearnonlinear 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==

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}}

*{{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 }}