Spiral states, first-order transitions and specific heat multipeak phenomenon in $J_1$-$J_2$-$J_3$ Ising model: A Wang-Landau algorithm study

The classical $J_1$-$J_2$-$J_3$ Ising model on the honeycomb lattice is important for understanding frustrated magnetic phenomena in materials such as FePS$_3$ and Ba$_2$CoTeO$_6$, where diverse phases (e.g., striped, zigzag, armchair) and magnetization plateaus have been experimentally observed. To explain the experimental results, previous mean-field studies have explored its thermal phase transitions, identifying armchair phases and striped phases, but their limitations call for more reliable numerical investigations. In this work, we systematically revisit the classical $J_1$-$J_2$-$J_3$ Ising model using the Wang-Landau algorithm. We find that the armchair (AC) phase, previously reported in mean-field and experimental studies, actually coexists with the spiral (SP) phase, with their combined degeneracy reaching 20-fold (4-fold for the AC states and 16-fold for the spiral states). The phase transitions and critical exponents are studied at different interaction values. We observe first-order phase transitions, continuous phase transitions, and even the multipeak phenomenon in frustrated systems. These results clarify the nature of phases and phase transitions in frustrated Ising systems and their exponents, and additionally provide inspiration for experimental efforts to search for the spiral state and specific-heat multipeak phenomenon.