Real space-time renormalization for the Luttinger-Thirring model on a lattice

We study a lattice version of the Luttinger-Thirring model. The Migdal and the one-hypercube approximations locate exactly the fixed point and give the eigenvalues correctly. For $d=2\mathrm{}$, a quartic operator is found to be marginal at the fixed point, in the Migdal approximation, and to be completely marginal, giving rise to a line of fixed points, in the one-hypercube approximation. Due to a rescaling of the fermion fields the Migdal approximation does not suffer from the usual trouble of working well only for systems with $\frac{\beta }{\nu }\simeq 0$. In the one-hypercube approximation the spin rescaling factor is determined variationally using Barber's variational Pontryagin principle and has, at the fixed point, the value it should have for a nontrivial fixed point to be reached. The backward scattering model is also studied within the Migdal approximation.