Finite-size scaling studies of massive one-dimensional lattice models

In this paper we propose an efficient routine to carry out finite size scaling studies of one-dimensional massive lattice models. Unlike massless systems, for which the lattice and continuum models yield the same results for low-lying excitations due to the infinite correlation length, the massive continuum models can at best approximate their lattice counterparts because of the intrinsic length scale xi similar to Delta -1, where Delta is the mass gap. On examples of antiferromagnetic gapped spin-1/2 XXZ chains we show explicitly that several relations between physical quantities like the mass gap, spin-wave velocity upsilon, and the correlation length xi, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when xi is comparable with the lattice spacing ao. On the other hand, we find if the dispersion for elementary excitations is modified from the "Lorentz invariant" form root Delta (2)+upsilon (2)p(2) to root Delta (2)+upsilon (2)sub(2)p, the spin-wave velocity upsilon becomes almost size independent, and the appropriately defined correlation length using that dispersion agrees very well with numerical results. In fact, this follows from the Bethe ansatz solution for the spin-1/2 XXZ chains. Based on our previous experience and these considerations we propose scaling equations to obtain the a-state energy, mass gap, correlation length, spin-wave velocity and scattering length between the massive elementary excitations. Only three low energy levels are needed in this method. Although the method is illustrated on the gapped XXZ quantum spins, it is applicable to all one-dimensional lattice models with massive relativistic low energy dispersions. To substantiate our statement we show explicitly that the numerical data for the Ising model in a transverse magnetic field fully agree with the finite-size scaling based on the correlation length newly defined from the exact analytic solution.