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Journal of Physics A Mathematical and General

Bethe-ansatz solutions of a non-string type: Numerical results

L.V. Avdeev

Published Jul 7, 1989• 2 citations
Publisher

Abstract

For the integrable XXX antiferromagnetic ring of N spins s=1 or s=1/2 the numerical solutions to the Bethe-ansatz equations are found, which involve non-string configurations, namely multiplets. The results up to sN=150 are compared with higher-level Bethe-ansatz predictions. The absolute difference between the predicted and finite-N energies of the spin-zero states with a multiplet and four holes is of O(1/N). The coefficient is not the same as for the vacuum and depends on the positions of the holes. As has been expected, the multiplets are of an exponential accuracy in N, while sea strings are much more strongly deformed.

DOI
10.1088/0305-4470/22/13/004

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For the integrable XXX antiferromagnetic ring of N spins s=1 or s=1/2 the numerical solutions to the Bethe-ansatz equations are found, which involve non-string configurations, namely multiplets. The results up to sN=150 are compared with higher-level Bethe-ansatz predictions. The absolute difference between the predicted and finite-N energies of the spin-zero states with a multiplet and four holes is of O(1/N). The coefficient is not the same as for the vacuum and depends on the positions of the holes. As has been expected, the multiplets are of an exponential accuracy in N, while sea strings are much more strongly deformed.
The Bethe-ansatz equations for the integrable spin-s isotropic Heisenberg antiferromagnet are solved numerically at a finite number of sites N. The vacuum and the lowest singlet and triplet solutions are presented for s=1/2, . . ., 2 up to sN=240. Through extrapolation of the finite-size data, the central charge and anomalous dimensions of the scaling operators for the underlying conformal field theory are calculated and found to agree with the expected theoretical values. The status of the Bethe string hypothesis about the structure of the solutions is discussed based on the obtained computer data up to s=9/2.
We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta knk_nkn​ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.
The Bethe ansatz provides an exact solution framework for integrable quantum spin chains, yet the detailed structure of its solutions remains subtle. In this work, we solve the Bethe ansatz with SU(N) global symmetry numerically from two perspectives. First, we systematically investigate approximate solutions known as strings, derived from the so-called string hypothesis. These solutions can be labeled by rigged configurations. Second, we compute physically admissible Bethe roots using the Wronskian Bethe equations, whose solutions admit a natural labeling by standard Young tableaux (SYT). These two labeling schemes, SYT and rigged configurations, arise independently in combinatorics. A key combinatorial result, the Kerov-Kirillov-Reshetikhin (KKR) bijection, establishes a one-to-one correspondence between them. While this bijection is well understood in a purely combinatorial setting, its role within the physics of integrable systems has not been systematically examined. To investigate the KKR bijection within integrable spin chains, we perform numerical computations of Bethe roots for su(N) chains up to length L=20. Remarkably, our numerical results show that the SYT labeling of solutions obtained from the Wronskian Bethe equations closely matches the rigged configuration labeling prescribed by the KKR bijection when compared using a nearest neighbor distance measure. Since the two labeling schemes were developed independently and, in particular, the SYT labeling does not rely on the string hypothesis, this correspondence reveals a deep structural relation between the combinatorial and analytic descriptions of integrable systems.
Bethe ansatz equations (BAE) are investigated for the isotropic spin- 1 / 2 Heisenberg chain as well as the XXZ chain with real solutions characterized by identical integers in the logarithmic form of the BAE. Such states do not belong to the usual classification scheme (string hypothesis). They always exist in critical sectors with respect to the total spin of the chain and the anisotropy. The conception of holes has to be changed in this case, because there occur more holes than expected. The finite-size corrections for the XXX model are calculated including complex roots and the new type of real solutions with repeating integers. The 'tower structure' predicted by the hypothesis of conformal invariance is shown to remain unchanged.
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)