  We discuss quantum field theories with global $SU(N)$ and $O(N)$ symmetries
for which the temporal direction is compactified on a circle of size $L$ with
periodicity of fields up to a global symmetry transformation, i.e. twisted
boundary conditions. Such boundary conditions correspond to an insertion of the
global symmetry operator in the partition function. We argue that for a special
choice of twists most of the excited states get projected out, leaving only
either mesonic states or states whose energy scales with $N$. When
$N\rightarrow \infty$ all excitations become suppressed at any compact radius
and the twisted partition function gets a contribution from the ground-state
only, rendering observables independent of the radius of compactification, i.e.
volume independent. We explicitly prove that this is indeed the case for the
$CP(N-1)$ and $O(N)$ non-linear sigma models in any number of dimensions. We
further focus on the two-dimensional $CP(N-1)$ case which is asymptotically
free, and demonstrate, unlike its thermal counterpart, the twisted theory has
commuting $N\rightarrow\infty,L\rightarrow\infty$ limits and does not undergo a
second-order phase transition at "zero-temperature" discussed by Affleck long
ago. At finite $L$ the theory is described by an effective, zero-temperature
quantum mechanics with smoothly varying parameters depending on $L$,
eliminating the possibility of a phase transition at any $L$, which was
conjectured by \"Unsal and Dunne. As $L$ is decreased at fixed and finite $N$
the relevant objects dictating the $\theta$ dependence are quantum
kink-instantons, avatars of the small $L$ regime fractional instantons. These
considerations, for the first time establishes the idea of adiabatic continuity
advocated by \"Unsal et. al.
