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Journal of Physics A Mathematical and General

Solution of the Bethe ansatz equations with complex roots for finite size: the spin S1 isotropic and anisotropic chains

Hector J. de Vega, Ferenc Woynarovich

Published May 7, 1990• 21 citations
Publisher

Abstract

The Bethe ansatz equations for spin-S (S>or=1) integrable vertex models (and magnetic chains) where the ground state is formed by complex roots are investigated for finite-size N. It is shown that the finite-size corrections to the imaginary parts of the roots (Bethe strings) for N>>1 are given by alpha m /(N sigma ( eta )) where eta is the real part of the roots, sigma ( eta ) is the density of the real parts, and m is the index of a root within a string. The constants alpha m are determined by a set of algebraic equations, and are given explicitly by alpha m =1/ pi ln cos(1/2 pi (S-m-1)/(S+1))/cos(1/2 pi (S-m)/(S+1)). For the best known, S=1, case alpha 0 =ln 2/(2 pi ). These results are found through a generalisation of the Euler-Maclaurin formula including nonanalytic contributions in N -1 which turn out to be essential in the solution of the present problem.

DOI
10.1088/0305-4470/23/9/022

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The Bethe ansatz equations for spin-S (S>or=1) integrable vertex models (and magnetic chains) where the ground state is formed by complex roots are investigated for finite-size N. It is shown that the finite-size corrections to the imaginary parts of the roots (Bethe strings) for N>>1 are given by alpha m /(N sigma ( eta )) where eta is the real part of the roots, sigma ( eta ) is the density of the real parts, and m is the index of a root within a string. The constants alpha m are determined by a set of algebraic equations, and are given explicitly by alpha m =1/ pi ln cos(1/2 pi (S-m-1)/(S+1))/cos(1/2 pi (S-m)/(S+1)). For the best known, S=1, case alpha 0 =ln 2/(2 pi ). These results are found through a generalisation of the Euler-Maclaurin formula including nonanalytic contributions in N -1 which turn out to be essential in the solution of the present problem.
It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.
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.
We determine the motions of the roots of the Bethe ansatz equation for the ground state in the XXZ spin chain under a varying twist angle. This is done by analytic as well as numerical study in a finite size system. In the attractive critical regime 0 < Δ < 1, we reveal intriguing motions of strings due to the finite size corrections to the length of the strings: in the case of two-strings, the roots collide into the branch points perpendicularly to the imaginary axis, while in the case of three-strings, they fluctuate around the center of the string. These are successfully generalized to the case of n-string. These results are used to determine the final configuration of the momenta as well as that of the phase shift functions. We obtain these as well as the period and the Berry phase in the regime Δ ≤ -1 also, establishing the continuity of the previous results at -1 < Δ < 0 to this regime. We argue that the Berry phase can be used as a measure of the statistics of the quasiparticle (or the bound state) involved in the process.
The spin-1/21/21/2 Heisenberg antiferromagnetic chain is the canonical example of an integrable quantum many-body model. Despite its exact solvability, explicit finite-size solutions are typically only accessible via numerical evaluation of the Bethe ansatz equations. Here, we analyse the algebraic structure of the exact, symbolic ground states for chains up to twelve sites using the Bethe ansatz. We find that the algebraic degree of the ground state magnon phases and Bethe roots scale with respect to system size at a different rate to the ground-state energy and wavefunction coefficients. The ground-state Bethe roots and energy are found to become Galois unsolvable for ten or more sites, thus demonstrating the lack of a symbolic analytical solution in a quantum integrable model due to algebraic complexity.
Abstract The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.
We determine the motions of the roots of the Bethe ansatz equation for the ground state in the XXZ spin chain under a varying twist angle. This is done by analytic as well as numerical study at a finite size system. In the attractive critical regime 0< \\Delta <1 , we reveal intriguing motions of strings due to the finite size corrections to the length of the strings: in the case of two strings, the roots collide into the branch points perpendicularly to the imaginary axis, while in the case of three strings, they fluctuate around the center of the string. These are successfully generalized to the case of n-string. These results are used to determine the final configuration of the momenta as well as that of the phase shift functions. We obtain these as well as the period and the Berry phase also in the regime \\Delta \\leq -1, establishing the continuity of the previous results at -1 < \\Delta < 0 to this regime. We argue that the Berry phase can be used as a measure of the statistics of the quasiparticle ( or the bound state) involved in the process.