Abstract We present a simple derivation of the Luttinger liquid relation for the one-dimensional Hubbard model, considering a finite U and in the U = ∞ limit. We describe a simple solution of the Hubbard model in the infinite repulsion limit and use it to calculate the correlators of the model in this limit in a simple and physical way using the bosonization technique. We then calculate the asymptotics of the correlators of the model at arbitrary U through a single parameter, which can be calculated from the Bethe Ansatz solution. Our derivation of the critical exponents is simple and allows one to express different physical operators of the Hubbard model through the charge and spin Bose fields in a direct and physically transparent way.
The one-dimensional Hubbard model is characterized by charge-spin separation, but nevertheless in the doped system \ensuremath{\sigma}(\ensuremath{\omega}>0) does not vanish due to a small remaining coupling between charge and spin degrees of freedom. We derive an effective Hamiltonian and give in leading order (\ensuremath{\sim}
J2) an analytic expression for \ensuremath{\sigma}(\ensuremath{\omega}) for the case of the Mott insulator doped with one hole. The complete frequency dependence in the strong-coupling limit of the Hubbard model and for the t-J model agrees with results from exact diagonalization. The limit \ensuremath{\sigma}(\ensuremath{\omega}\ensuremath{\rightarrow}0) at T=0 is found to be very different for the two models, i.e., \ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{3/2}$ in the strong-coupling limit and \ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}1/2}$ for the t-J model.