  The ordering temperature of a quasi-one-dimensional system, consisting of
weakly interacting quantum spin-1/2 chains with antiferromagnetic
spin-frustrating couplings (or zig-zag ladder) is calculated. The results show
that a quantum critical point between two phases of the one-dimensional
subsystem plays a crucial role. If the one-dimensional subsystem is in the
antiferromagnetic-like phase in the ground state, similar to the phase of a
spin chain without frustration, weak couplings yield magnetic ordering of the
Neel type. For intra-chain spin-frustrating interactions larger than the
critical one (at which the quantum phase transition takes place), the
quasi-one-dimensional spin system manifests a spiral magnetic incommensurate
ordering. The obtained results of our quantum theory are compared with the
quasi-classical approximations. The calculated features of magnetic ordering
are expected to be generic for weakly coupled quantum spin chains with gapless
excitations and spin-frustrating nearest and next-nearest neighbor
interactions.
