Complete Classification of Domain Wall Solutions in the $\mathbb{Z}_2$-symmetric 2HDM

We present a complete classification of domain wall solutions in the two-Higgs Doublet Model (2HDM) with a global $\mathbb{Z}_2$ symmetry, categorised as superconducting, CP-violating, or neither, depending on the scalar particle masses and the ratio of the two Higgs doublets' vacuum expectation values. We demonstrate that any domain wall solution can be reduced to depend on only six of the eight general field components, with further field reductions possible within different regions of the parameter space. Furthermore, we show that the superconducting solutions can be used to construct stable, current-carrying domain walls in two spatial dimensions. Similarly, the CP-violating solutions allow for two-dimensional configurations where CP symmetry is locally broken on the $\mathbb{Z}_2$-symmetric wall, which could provide an out-of-equilibrium environment for CP-violating processes to occur.