\@writefile{lof}{\contentsline{figure}{\numberline{1}{\ignorespaces$\Delta\tau t $ extrapolation for the 4-site Hubbard ring. }}{2}{figure.1}}

\@writefile{lof}{\contentsline{figure}{\numberline{2}{\ignorespaces Spin correlation functions along one leg for the Hubbard ladder. As $t_y$ grows the spin gap becomes large enough so as to detect the exponential decal of the spin correlation function on this small lattice size. The underlying physics of odd-even ladder systems is introduced in the article: Elbio Dagotto and T. M. Rice, Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials, Science 271 (1996), no. 5249, 618?623. }}{5}{figure.2}}

\newlabel{Ladder.fig}{{2}{5}{Spin correlation functions along one leg for the Hubbard ladder. As $t_y$ grows the spin gap becomes large enough so as to detect the exponential decal of the spin correlation function on this small lattice size. The underlying physics of odd-even ladder systems is introduced in the article: Elbio Dagotto and T. M. Rice, Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials, Science 271 (1996), no. 5249, 618?623}{figure.2}{}}

\@writefile{lof}{\contentsline{figure}{\numberline{3}{\ignorespaces Density-Density correlation functions of the t-V model. In the Luttinger liquid phase, $-2 < V/t < 2$ it is known that the density -density correlations decay as $\delimiter "426830A n(r) n(0)\delimiter "526930B \propto\qopname\relax o{cos}(\pi r) r^{-\left(1+K_\rho\right)}$ with $\left(1+K_\rho\right)^{-1}=\frac{1}{2}+\frac{1}{\pi}\qopname\relax o{arcsin}\left(\frac{V}{2 | t | }\right)$ (A. Luther and I. Peschel, Calculation of critical exponents in two dimensions from quantum field theory in one dimension, Phys. Rev. B 12 (1975), 3908.) The interested reader can try to reproduce this result. }}{6}{figure.3}}

\newlabel{tV.fig}{{3}{6}{Density-Density correlation functions of the t-V model. In the Luttinger liquid phase, $-2 < V/t < 2$ it is known that the density -density correlations decay as $\langle n(r) n(0)\rangle\propto\cos(\pi r) r^{-\left(1+K_\rho\right)}$ with $\left(1+K_\rho\right)^{-1}=\frac{1}{2}+\frac{1}{\pi}\arcsin\left(\frac{V}{2 | t | }\right)$ (A. Luther and I. Peschel, Calculation of critical exponents in two dimensions from quantum field theory in one dimension, Phys. Rev. B 12 (1975), 3908.) The interested reader can try to reproduce this result}{figure.3}{}}