Properties of the zigzag spin chains with various nearest-neighbor and next-nearest-neighbor interactions are studied by making use of the transfer-matrix renormalization group method. Thermodynamic quantities of the systems (temperature dependence of the susceptibility and the specific heat), as well as the field dependence of the magnetization are analyzed numerically with a high accuracy in the thermodynamic limit. The results have been compared with the recent experimental data on Rb2Cu2Mo3O12.

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The temperature dependence of the specific heat and spin susceptibility of the spin ladders with cyclic four-spin interactions in the rung-singlet phase is explored by making use of the transfermatrix renormalization group method. The values of spin gap are extracted from the specific heat and susceptibility, respectively. It is found that for different relative strength between interchain and intrachain interactions, the spin gap is approximately linear with the cyclic four-spin interaction in the region far away from the critical point. Furthermore, we show that the dispersion for the one-triplet magnon branch can be obtained by numerically fitting on the partition function.