Domain State of the ANNNI model in Two Dimensions

We have examined the spin ordering of an axial next-nearest-neighbor Ising (ANNNI) model in two dimensions (2D) near above the antiphase ($\langle 2 \rangle$ phase). We considered an $N_R$-replica system and calculated an overlap function $q_m$ between different replicas having used a cluster heat bath (CHB) Monte Carlo method. We determined transition temperature between $\langle 2 \rangle$ phase and a floating incommensurate (IC) phase as $T_{C2}/J = 0.89 \pm 0.01$ with frustration ratio $\kappa (\equiv -J_2/J_1) = 0.6$. We found that the spin state at $T \gtrsim T_{C2}$ may be called as a domain state, because the spin structure is characterized by a sequentially arranged four types of domains with different $\langle 2 \rangle$ structures. In the domain state, the 2D XY symmetry of the spin correlation in the IC phase weakly breaks and the diversity of the spin arrangement increases as $T \rightarrow T_{C2}$. The Binder ratio $g_L$ exhibits a depression at $T \sim T_{C2}$ and the quasi-periodic spin structure, which is realized in the IC phase, becomes diverse at $T \gtrsim T_{C2}$. We discussed that the domain state is stable against the thermal fluctuation which brings a two-stage development of the spin structure at low temperatures.