  We study the dependence of the ground state energy on an applied
Aharonov-Bohm flux $\Phi$ for the Luttinger model with large momentum
scattering. Employing the method of finite size bosonization, we show that for
systems with a spin gap but with gapless charge degrees of freedom, the ground
state energy has an exact period of $hc/2e$, i. e. {\em half} a flux quantum,
in the limit of large system size $L$. Finite size corrections are found to
vanish exponentially in $L$. This behavior is contrasted to that of the spin
gapless case, for both even and odd particle number. Generalizations to finite
temperature are also discussed.
