  We perform a systematic investigation of an asymmetric zig-zag spin ladder
with inter-leg exchange $J_1$ and different exchange integrals $J_2 \pm \delta$
on both legs. In the weak limit of frustration, the spin model can be mapped to
a revised double frequency Sine-Gorden model by using bosonization.
Renormalization group analysis shows that the Heisenberg critical point flows
to an intermediate-coupling fixed point with gapless excitations and a
vanishing spin velocity. When the frustration is large, a spin gap opens and a
dimer liquid is realized. Fixing $J_2 = J_1 /2$, we find, as a function of
$\delta$, a continuous manifold of Hamiltonians with dimer product ground
states, interpolating between the Majumdar-Ghosh and sawtooth spin-chain model.
While the ground state is independent of the alternating next-nearest-neighbor
exchange $\delta$, the gap size of excitations is found to decrease with
increasing $\delta$. We also extend our study to a two-dimensional double layer
model with an exactly known ground state.
