We consider here the XXZ spin chain perturbed by the operator ${\ensuremath{\sigma}}^{x}$ (``in a transverse field'') which is a lattice regularization of the sine-Gordon model. This can be shown using conformal perturbation theory. We calculate the mass ratios of particles which lie in a discrete part of the spectrum and obtain results in accord with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states (UV or conformal limit). Our result for conformal dimension of the second breather state is different from that conjectured by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)] and is consistent with this paper for other states.
We consider here the massive Thirring model regularized with the
XYZ spin chain. We numerically calculate the mass ratios of particles which lie in the discrete part of the spectrum and obtain results in accordance with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states and evaluate the conformal dimensions. This paper, taken together with the previous one for the sine-Gordon model, confirms the duality relation between two models formulated by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)].
With the use of the Matsubara technique three things are done: (1) the lifetime of the
n=1 breather (i.e., the renormalized phonon) is determined, (2)
(2M1)M2 in the weak-coupling limit is calculated, and (3)
(2Es)M1 near ${g}^{2}=4\ensuremath{\pi}$ is determined by making use of the equivalence between the sine-Gordon model and the massive Thirring model. Here
Mn is the mass of the
nth breather and
Es is the soliton energy. It is also shown that the ${\ensuremath{\varphi}}^{4}$ model has the identical bosonic spectrum to that of the sine-Gordon system in the weak-coupling limit and in the low-temperature limit.
In the quantum inverse scattering method, the sine-Gordon model is solved on a lattice, and its continuum and infinite-volume limits become uniform only after the length of the box is rescaled. We show that this rescaling, supplemented by an on-shell condition, defines a renormalization scheme. The whole dependence on the cut-off is absorbed into a factor
ZL multiplying the length of the box, with an exponent fixed by the scaling dimension of the vertex operator; neither the field nor the coupling
β is renormalized, no subtraction scale is introduced, and the mass parameter receives a finite renormalization, fixed by requiring that the lightest breather has the mass of the boson in the linearized theory. The soliton and breather masses follow from the eigenvalues of the monodromy operator as functions of parameters that do not run, and reduce to the classical and semiclassical results as
β→0. In particular, the mass ratios are produced directly from the regularized construction, while the relation between the lattice parameters and the physical scale is absorbed into
ZL and never needs to be computed. The scheme is a reparameterization of the theory which makes explicit the interplay between integrability and renormalization. We compare it with Coleman's normal ordering and with conformal perturbation theory, and we argue that it extends to ultralocal integrable theories with a massive spectrum, possibly including asymptotically free models that admit an ultralocal lattice regularization.