Delete ALF_Tutorial.aux

Tutorial/ALF_Tutorial.aux

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-\@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}{}}