Metastable states in the triangular-lattice Ising model studied by Monte Carlo simulations: Application to the spin-chain compound<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mtext>Ca</mml:mtext></mml:mrow><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mrow><mml:mtext>Co</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mtext>O</mml:mtext><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>

It is well known that the spin-chain compound ${\text{Ca}}_{3}{\text{Co}}_{2}{\text{O}}_{6}$ exhibits interesting plateaus in the magnetization as a function of the magnetic field at low temperatures. The origin of them is still controversial. In this paper, we study the thermal behavior of this compound with a single-flip Monte Carlo simulation on a triangular lattice and demonstrate the decisive influence of metastable states on the splitting of the ferrimagnetic $1/3$ plateau below 10 K. We consider the ${[{\text{Co}}_{2}{\text{O}}_{6}]}_{n}$ chains as giant magnetic moments described by large Ising spins on planar clusters with open boundary conditions. With this simple frozen-moments model we obtain stepped magnetization curves which agree quite well with the experimental results for different sweeping rates. We describe particularly the out-of-equilibrium states that split the low-temperature $1/3$ plateau into three steps. They relax thermally to the $1/3$ plateau, which has long-range order at equilibrium. Such metastable states are further analyzed with snapshots unveiling an interlinked mobile domain walls structure that is responsible for the observed behavior of the $1/3$ plateau. A comparison is also given of our classical Monte Carlo results with exact diagonalization results in small triangular quantum clusters, providing further support for our thermal description of this compound.