The nature of the ordered phases and the character of the phase transitions of the Ising chain model with intrachain interaction $j$, coupled by interchain direct $j_2$, indirect $j_1$, and four-spin $k$ interactions, are studied using a hybrid method—a combination of the transfer matrix (TM) and molecular field approximation (MFA). Indirect coupling is achieved through redundant spins that can be rigorously traced out in the partition function, which leads to a temperature-dependent resultant (effective) interchain interaction. In this approach, an exactly solvable one-dimensional model is placed in the molecular field originating from adjacent chains. The temperature dependences of the magnetization, specific heat, and correlation functions in the ordered phases are found. The phase diagram in the plane (transition temperature $t_c, k$) with three ordered phases— $FM$ ferromagnetic, $AF$ antiferromagnetic, $M$ with alternating ferro- and antiferromagnetic chains—and a re-entrant paramagnetic phase has been determined. It has been shown that the four-spin interaction $k$ can change the phase transition kind from continuous to discontinuous and lead to a gigantic specific heat at a temperature above $t_c$.

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• Classical statistical mechanics
• Order parameters
• Phase diagrams
• Phase transitions