Luttinger-liquid Parameter of Hubbard Chain and Hubbard Ladder

We study the Luttinger-liquid parameter $K_{\rho}$ of the Hubbard chain and the Hubbard ladder models by the ordinary perturbation method combined with the Luttinger-liquid relation. According to the Luttinger-liquid relation, the critical exponent $K_{\rho}$ is related to the charge susceptibility $\chi_c$ and the Drude weight D by $K\sb{\rho}={1/2}(\pi \chi_c D)^{1/2}$. By calculating these quantities with the perturbation method, we obtain $K_{\rho}$ at the first-order analytically and up to the second-order numerically. We compare these results with results of the Bethe ansatz for the Hubbard chain and that of the numerical diagonalization for the Hubbard ladder. It shows that the perturbation calculation of $K_{\rho}$ is reliable in the weak coupling regime.