Bethe strings in the dynamical structure factor of the spin-1/2 Heisenberg XXX chain

Abstract

Recently there has been a renewed interest in the spectra and role in dynamical properties of excited states of the spin-1/2 Heisenberg antiferromagnetic chain in longitudinal magnetic fields associated with Bethe strings. The latter are bound states of elementary magnetic excitations described by Bethe-ansatz complex non-real rapidities. Previous studies on this problem referred to finite-size systems. Here we consider the thermodynamic limit and study it for the isotropic spin-1/2 Heisenberg XXX chain in a longitudinal magnetic field. We confirm that also in that limit the most significant spectral weight contribution from Bethe strings leads to (k, omega)-plane gapped continua in the spectra of the spin dynamical structure factors S+-(k, omega) and S-xx(k, omega) = S-yy(k, omega). The contribution of Bethe strings to S-zz(k, omega) is found to be small at low spin densities m and to become negligible upon increasing that density above m approximate to 0.317. For S-+(k, omega), that contribution is found to be negligible at finite magnetic field. We derive analytical expressions for the line shapes of S+-(k, omega), S-xx(k, omega) = S-yy(k, omega), and S-zz(k, omega) valid in the (k, omega)-plane vicinity of singularities located at and just above the gapped lower thresholds of the Bethe-string states's spectra. As a side result and in order to provide an overall physical picture that includes the relative (k, omega)-plane location of all spectra with a significant amount of spectral weight, we revisit the general problem of the line-shape of the transverse and longitudinal spin dynamical structure factors at finite magnetic field and excitation energies in the (k, omega)-plane vicinity of other singularities. This includes those located at and just above the lower thresholds of the spectra that stem from excited states described by only real Bethe-ansatz rapidities. (C) 2020 The Author(s). Published by Elsevier B.V.