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arXiv:1402.0651 · Journal of Physics A Mathematical and Theoretical

Singular solutions to the Bethe ansatz equations and rigged configurations

Anatol N. Kirillov, Reiho Sakamoto

Published May 6, 2014• 24 citations
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Abstract

We provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.

DOI
10.1088/1751-8113/47/20/205207

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We provide a conjecture for the following two quantities related to the spin- isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
The Bethe equations for the isotropic periodic spin-1/2 Heisenberg chain with N sites have solutions containing ±i/2 that are singular: both the corresponding energy and the algebraic Bethe ansatz vector are divergent. Such solutions must be carefully regularized. We consider a regularization involving a parameter that can be determined using a generalization of the Bethe equations. These generalized Bethe equations provide a practical way of determining which singular solutions correspond to eigenvectors of the model.
Abstract Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = ½ (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations – magnetic configurations which are just possible distributions of spins over the nodes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N = 12, where the central rapidity of some of the odd-length strings becomes complex so that not all the strings self-conjugate individually. We show that there are at most ( N − 2)/2 singular solutions for M = 4, M = 5 down-spins and at most ( N 2 − 6 N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ⩾ 2 M . Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.