Dynamical structure factor at smallq for theXXZ spin-1/2 chain

Abstract

We combine Bethe ansatz and field theory methods to study the longitudinal dynamical structure factorSzz(q,ω) for theanisotropic spin-1/2 chain in the gapless regime. Using bosonization, we derive a low-energy effectivemodel, including the leading irrelevant operators (band curvature terms) whichaccount for boson decay processes. The coupling constants of the effectivemodel for finite anisotropy and finite magnetic field are determined exactly bycomparison with corrections to thermodynamic quantities calculated by Betheansatz. We show that a good approximation for the shape of the on-shell peak ofSzz(q,ω) inthe interacting case is obtained by rescaling the result for free fermions by certain coefficients extractedfrom the effective Hamiltonian. In particular, the width of the on-shell peak is argued to scale likeδωq∼q2 and this prediction is shown to agree with the width of the two-particle continuumat finite fields calculated from the Bethe ansatz equations. An exception to theq2 scaling is found at finite field and large anisotropy parameter (near theisotropic point). We also present the calculation of the high-frequency tail ofSzz(q,ω) in the region using finite-order perturbation theory in the band curvature terms. Both thewidth of the on-shell peak and the high-frequency tail are compared withSzz(q,ω) calculated by Bethe ansatz for finite chains using determinant expressions for the formfactors and excellent agreement is obtained. Finally, the accuracy of the form factors ischecked against the exact first moment sum rule and the static structure factor calculatedby density matrix renormalization group (DMRG).