In this work, we investigate the phase transitions and critical behaviors of the frustrated J1-J2-J3 Ising model on the square lattice using Monte Carlo simulations, and particular attention goes to the effect of the second next nearest neighbor interaction J3 on the phase transition from a disordered state to the single stripe antiferromagnetic state. A continuous Ashkin-Teller-like transition behavior in a certain range of J3 is identified, while the 4-state Potts-critical end point [J3/J1]C is estimated based on the analytic method reported in earlier work [Jin et al., Phys. Rev. Lett. 108, 045702 (2012)]. It is suggested that the interaction J3 can tune the transition temperature and in turn modulate the critical behaviors of the frustrated model. Furthermore, it is revealed that an antiferromagnetic J3 can stabilize the staggered dimer state via a phase transition of strong first-order character.
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