  The interplay between Kondo effect, indirect magnetic interaction and
geometrical frustration is studied in the Kondo lattice on the one-dimensional
zigzag ladder. Using the density-matrix renormalization group (DMRG), the
ground state and various short- and long-range spin- and density-correlation
functions are calculated for the model at half-filling as a function of the
antiferromagnetic Kondo interaction down to $J=0.3t$ where $t$ is the
nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown
to lead to at least two critical points: Starting from the strong-$J$ limit,
where almost local Kondo screening dominates and where the system is a
nonmagnetic Kondo insulator, antiferromagnetic correlations between
nearest-neighbor and next-nearest-neighbor local spins become stronger and
stronger, until at $J^{\rm dim}_{\rm c} \approx 0.89t$ frustration is
alleviated by a spontaneous breaking of translational symmetry and a
corresponding transition to a dimerized state. This is characterized by
antiferromagnetic correlations along the legs and by alternating antiferro- and
ferromagnetic correlations on the rungs of the ladder. A mechanism of partial
Kondo screening that has been suggested for the Kondo lattice on the
two-dimensional triangular lattice is not realized in the one-dimensional case.
Furthermore, within the symmetry-broken dimerized state, there is a magnetic
transition to a $90^{\circ}$ quantum spin spiral with quasi-long-range order at
$J^{\rm mag}_{\rm c} \approx 0.84t$. The quantum-critical point is
characterized by a closure of the spin gap (with decreasing $J$) and a
divergence of the spin-correlation length and of the spin-structure factor
$S(q)$ at wave vector $q=\pi/2$. This is opposed to the model on the
one-dimensional bipartite chain, which is known to have a finite spin gap for
all $J>0$ at half-filling.
