  We propose an analytical ansatz, using which the ordering temperature of a
quasi-one-dimensional (quasi-1D) antiferromagnetic (AF) system (weakly coupled
quantum spin-1/2 chains) in the presence of the external magnetic field is
calculated. The field dependence of the critical exponents for correlation
functions of 1D subsystems plays a very important role. It determines the
region of possible re-entrant phase transition, governed by the field. It is
shown how the quantum critical point between two phases of the 1D subsystem,
caused by spin-frustrating next-nearest neighbor (NNN) and multi-spin ring-like
exchanges, affects the field dependence of the ordering temperature. Our
results qualitatively agree with the features, observed in experiments on
quasi-1D AF systems.
