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 $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}1},$ where $\ensuremath{\Delta}$ 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
v, and the correlation length \ensuremath{\xi}, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when $\ensuremath{\xi}$ is comparable with the lattice spacing
a0. On the other hand, we find if the dispersion for elementary excitations is modified from the ``Lorentz invariant'' form $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{p}^{2}}$ to $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{\mathrm{sin}}^{2}p},$ the spin-wave velocity
v 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 ground-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.
We consider the transverse dynamical structure factor of the anisotropic Heisenberg spin-1/2 chain ( XXZ model) in the gapped antiferromagnetic regime (Δ>1). Specializing to the case of zero field, we use two independent approaches based on integrability (one valid for finite size, the other for the infinite lattice) to obtain the exact two-spinon part of this correlator. We discuss in particular its asymmetry with respect to the π/2-momentum line, its overall anisotropy dependence, and its contribution to sum rules.
We study the magnetization of gapped spin 1/2 XXZ Heisenberg Ising chains and calculate the scattering length among massive spinons. We obtain the magnetization close to the critical external magnetic field. The leading correction term determined by the scattering length among massive spinons is given. Our results are in agreement with exact results and with experimental results. We show that the deviation from the massive free hard core boson picture can be accounted very well by the leading correction term due to the spinon-spinon interaction. We show that the deviation increases along with the increasing of the Ising term.
We discuss a universal relation $\ensuremath{\varepsilon}(i\ensuremath{\kappa})=0$ with $\mathrm{Re}\ensuremath{\kappa}=1/\ensuremath{\xi}$ in 1D quantum spin systems with an excitation gap, where $\ensuremath{\varepsilon}(k)$ is the dispersion curve of the low-energy excitation and $\ensuremath{\xi}$ is the correlation length of the ground state. We first discuss this relation for integrable models such as the Ising model in a transverse filed and the
XYZ model. We secondly make a derivation of the relation for general cases, in connection with the equilibrium crystal shape in the corresponding 2D classical system. We finally verify the relation for the
S=1 bilinear-biquadratic spin chain and the
S=1/2 zigzag spin ladder numerically.
We calculate the ground-state energy and correlation length of the spin-1 antiferromagnetic biquadratic chain, as well as the excitation energy of the two lowest-lying states. We use the equivalence of this model to the spin-1/2 XXZ chain for a specific anisotropy parameter. The XXZ model is solvable by the Bethe ansatz. We solve the Bethe ansatz equations numerically and obtain the correlation length from the ground-state energy of finite chains by use of a finite-size scaling ansatz. We find a surprisingly large correlation length. We also perform quantum Monte Carlo simulations which show clearly that the biquadratic chain dimerizes.