The role of q-spin singlet pairs of physical spins in the dynamical properties of the spin-1/2 Heisenberg-Ising XXZ chain

Abstract

Dynamical correlation functions contain important physical information on correlated spin models. Here a dynamical theory suitable to the Δ=1 isotropic spin-1/2 Heisenberg chain in a longitudinal magnetic field is extended to anisotropy Δ>1. The aim of this paper is the study of the (k,ω)-plane line shape of the spin dynamical structure factor components S+−(k,ω), S−+(k,ω), and Szz(k,ω) of the Δ>1 spin-1/2 Heisenberg-Ising chain in a longitudinal magnetic field near their (k,ω)-plane sharp peaks. However, the extension of the theory to anisotropy Δ>1 requires as a first step the clarification of the nature of a specific type of elementary magnetic configurations in terms of physical spins 1/2. To reach that goal, that the spin SU(2) symmetry of the isotropic Δ=1 point is in the case of anisotropy Δ>1 replaced by a continuous quantum group deformed q-spin SUq(2) symmetry plays a key role. For Δ>1, spin projection Sz remains a good quantum number whereas spin S is not, being replaced by the q-spin Sq in the eigenvalue of the Casimir generator of the continuous SUq(2) symmetry. Based on the isomorphism between the irreducible representations of the Δ=1 spin SU(2) symmetry and Δ>1 continuous SUq(2) symmetry and on their relation to the occupancy configurations of the Bethe-ansatz quantum numbers one finds that the elementary magnetic configurations under study are q-spin neutral. This determines the form of S matrices on which the extended dynamical theory relies. They are found in this paper to describe the scattering of n-particles whose relation to configurations of physical spins 1/2 is established. Specifically, their internal degrees of freedom refer to unbound q-spin singlet pairs of physical spins 1/2 described by n=1 real single Bethe rapidities and to n=2,3,... bound such q-spin singlet pairs described by Bethe n-strings for n>1. There is a relationship between the negativity and the length of the momentum k interval of the k dependent exponents that control the power-law line shape of the spin dynamical structure factor components near the lower threshold of a given (k,ω)-plane continuum and the amount of spectral weight over the latter. Using such a relationship, one finds that in the thermodynamic limit the significant spectral weight contributions from Bethe n-strings at a finite longitudinal magnetic field refer to specific two-parametric (k,ω)-plane gapped continua in the spectra of the spin dynamical structure factor components S+−(k,ω) and Szz(k,ω). In contrast to the Δ=1 isotropic chain, excited energy eigenstates including up to n=3 Bethe n-strings lead to finite spectral-weight contributions to S+−(k,ω). Most spectral weight stems though from excited energy eigenstates whose q-spin singlet Sz=Sq=0 pairs of physical spins 1/2 are all unbound. It is associated with (k,ω)-plane continua in the spectra of the spin dynamical structure factor components that are gapless at some specific momentum values. We derive analytical expressions for the line shapes of S+−(k,ω), S−+(k,ω), and Szz(k,ω) valid in the vicinity of (k,ω)-plane lines of sharp peaks. Those are mostly located at and just above lower thresholds of (k,ω)-plane continua associated with both states with only unbound q-spin singlet pairs and states populated by such pairs and a single n=2 or n=3 n-particle. Our results provide physically interesting and important information on the microscopic processes that determine the dynamical properties of the non-perturbative spin-1/2 Heisenberg-Ising chain in a longitudinal magnetic field.