  We derive the lattice $\beta$-function for quantum spin chains, suitable for
relating finite temperature Monte Carlo data to the zero temperature fixed
points of the continuum nonlinear sigma model. Our main result is that the
asymptotic freedom of this lattice $\beta$-function is responsible for the
nonintegrable singularity in $\theta$, that prevents analytic continuation
between $\theta=0$ and $\theta=\pi$.
