Finite-time cooling and accessibility of the stripe phase in the Ising antiferromagnet

The finite-rate cooling dynamics of the triangular-lattice $J_1-J_2$ Ising antiferromagnet is studied under local Metropolis updates. Although an antiferromagnetic next-nearest-neighbor coupling selects a stripe phase in equilibrium, the simulations show that this phase is not automatically reached on finite time scales. A kinetic stripe-formation time $n^*(L,J_2/J_1)$ is defined from the probability of obtaining a globally stripe-ordered final state. This time shifts to much slower cooling as the system size increases and to faster cooling as $J_2/J_1$ increases. The size dependence is compatible with a coarsening-controlled process, with an effective growth at least quadratic in L over the simulated range. Real-space morphology and fixed-temperature diagnostics show that failed trajectories are not simply disordered states: they often contain locally stripe-ordered domains separated by residual walls or competing orientations. In the weak-$J_2/J_1$ regime, the system can restore the local nearest-neighbor frustrated constraint while still failing to select a global stripe sector. These results separate three processes that are usually conflated: energetic degeneracy lifting by $J_2/J_1$, local constraint restoration, and global stripe-orientation selection under local dynamics.