Abstract
In the previous chapter we have studied the consequences of integrability for the scattering theory. We saw that integrability implies a special form of the asymptotic wave function and factorisation of the multi-body scattering matrix. Our considerations were done for one-dimensional quantum-mechanical systems defined on an infinite line. To study thermodynamics, one needs, however, to confine the system in a finite volume.
A 1931 result that lay in obscurity for decades, Bethe’s solution to a quantum mechanical model now finds its way into everything from superconductors to string theory.
Murray T. Batchelor
The Bethe ansatz after 75 years,
Physics Today, January 2007
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Notes
- 1.
As an example, consider the \( N =4\) case and the operators \(T_2=S_{32}S_{42}S_{12}\) and \(T_3=S_{43}S_{13}S_{23}\). We have
$$\begin{aligned} T_2T_3= & {} S_{32}S_{42}S_{12}\cdot S_{43}S_{13}S_{23}=\underbrace{S_{32}S_{42}S_{43} }_\mathrm{YB} \cdot \underbrace{ S_{12}S_{13}S_{23} }_\mathrm{YB}= S_{43}S_{42}\underbrace{S_{32}\cdot S_{23}}_{=\mathbbm {1}}S_{13}S_{12}\\= & {} S_{43}S_{42}\cdot S_{13}S_{12}=S_{43}S_{13}\cdot S_{42}S_{12}=S_{43}S_{13}\underbrace{S_{23}\cdot S_{32}}_{=\mathbbm {1}}S_{42}S_{12}= T_3T_2\, . \end{aligned}$$ - 2.
This could be understood from the fact that to minimise energy for N even, one of the momenta must take zero value, so that the odd number of remaining integers \({\mathcal {I}}_j\) is distributed asymmetrically and maximally close to zero in one of two possible ways.
- 3.
This can be easily derived from the determinant formula (4.89).
- 4.
This structure is an implementation of the Schur-Weyl duality.
- 5.
That is the solutions which are not related to each other by permutations of some Bethe roots.
- 6.
The reader undoubtedly noticed a striking similarity of this expression to the Bethe wave function (4.100) in the fundamental sector. The “particle” coordinates are now positive integers that form an ordered set. The formula (5.70) constitute the Bethe hypothesis which was the starting point to obtain the solution of the delta-interaction Bose gas.
- 7.
- 8.
Compared to (5.28) we took the product of the Lax operators in an opposite order which is more convenient for our present treatment.
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Arutyunov, G. (2019). Bethe Ansatz. In: Elements of Classical and Quantum Integrable Systems . UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-24198-8_5
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