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Journal of Physics Communications

Observations on the Bethe ansatz solutions of the spin-1/2 isotropic anti-ferromagnetic Heisenberg chain

Bernie G. Nickel

Published Jan 30, 2019• 1 citation
Publisher

Abstract

Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.

DOI
10.1088/2399-6528/ab030b

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Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.
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 Calculations of low lying excited states in even length L → ∞ chains begun in 2019 J. Phys. Commun. 3 025007 in the total spin S = 0 sector are here extended to S > 0. Bethe string configurations that were observed to transition from their definition at small L to different ‘apparent’ configurations at large L are now understood to be a manifestation of the approximate chiral symmetry that develops with increasing chain length. It is conjectured that when chiral symmetry is built into the configuration definitions a complete explicit labelling of all low lying excited states by strings is obtained. The structure is of the form of a product of Bethe strings defining left and right chiral states of spin sL and sR and a central complex ‘string’ necessary to bind <?CDATA s⃗L{\vec{{\rm{s}}}}_{L}sL​?> s ⃗ L and <?CDATA s⃗R{\vec{{\rm{s}}}}_{R}sR​?> s ⃗ R into total <?CDATA S⃗.\vec{S}.S.?> S ⃗ . Rules based on the string content of a state yield its momentum and asymptotic energy, including the amplitude of the leading ln(L) correction.
The authors analyse the Bethe ansatz equations for the two-particle sector of the spin 1/2 Heisenberg XXX model on a one-dimensional lattice of length N. They show that, beginning at a critical lattice length of N=21.86, new pairs of real solutions develop, whereas complex solutions start to disappear. The integers (that appear in the logarithmic form of the Bethe equations) of the new solutions do not fit into the conventional classification scheme. The total number of solutions in the two-particle sector remains unchanged and is in agreement with the claim that the SU(2) extended Bethe ansatz gives a complete set of 2 N eigenstates.
Solutions to the Bethe Ansatz equations for the ground state of spin-½ isotropic anti-ferromagnetic periodic Heisenberg chains to length L = 2 21 are obtained by combining Lagrange interpolation with Newton–Raphson iteration. The long chain lengths allow many powers of a renormalization group running coupling constant to be included in fits to the ground state energy and make possible the confirmation of the convergence of the leading logarithmic term. The amplitude of this term is consistent with that expected on the basis of conformal field theory and the connection of the discrete spin-½ system to the continuum Wess–Zumino–Witten model. This resolves a decades old discrepancy based on analysis of shorter chains. An analytical improvement to the Hulthén wave-vector distribution is also provided.
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.