Quantum inverse scattering method: Difference between revisions

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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>.

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}}</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:

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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]].

==Main steps of the QISM==

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

# Take an [[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 relations

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

# Find correlators.

==References==

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*{{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]]