Magnetic field-induced deformation of the spin-density wave microphases in Ca$_3$Co$_2$O$_6$

The frustrated triangular Ising magnet Ca$_3$Co$_2$O$_6$ has long been known for an intriguing combination of extremely slow spin dynamics and peculiar magnetic orders, such as the evenly-spaced non-equilibrium metamagnetic magnetization steps and the long-wavelength spin density wave (SDW) order, the latter of which is essentially an emergent crystal of solitons. Recently, an elaborate field-cooling protocol to bypass the low-field SDW phase was proposed to overcome the extraordinarily long timescale of spin relaxation that impeded previous experimental studies in equilibrium, which may point to a deep connection between the low-temperature slow relaxation and the cooling process passing through the low-field SDW phase. As the first step to elucidate the conjectured connection, we investigate the magnetic field-induced deformation of the SDW state and incommensurate-commensurate transitions, thereby mapping out the equilibrium in-field phase diagram for a realistic three-dimensional lattice spin model by using Monte Carlo simulations. We also discuss Ginzburg-Landau theory that includes several Umklapp terms as well as an effective sine-Gordon model, which can qualitatively explain the observed magnetic field-induced deformation of the SDW microphases.