  We propose a cold atom implementation to attain the continuum limit of
(1+1)-d CP(N-1) quantum field theories. These theories share important features
with (3+1)-d QCD, such as asymptotic freedom and $\theta$ vacua. Moreover,
their continuum limit can be accessed via the mechanism of dimensional
reduction. In our scheme, the CP(N-1) degrees of freedom emerge at low energies
from a ladder system of SU(N) quantum spins, where the N spin states are
embodied by the nuclear Zeeman states of alkaline-earth atoms, trapped in an
optical lattice. Based on Monte Carlo results, we establish that the continuum
limit can be demonstrated by an atomic quantum simulation by employing the
feature of asymptotic freedom. We discuss a protocol for the adiabatic state
preparation of the ground state of the system, the real-time evolution of a
false $\theta$-vacuum state after a quench, and we propose experiments to
unravel the phase diagram at non-zero density.
