Classification of Abelian Domain Walls

We discuss domain walls from spontaneous breaking of Abelian discrete symmetries Z(N). A series of different domain wall structures are predicted, depending on the symmetry and charge assignments of scalars leading to the spontaneous symmetry breaking (SSB). A widely existing type of domain walls are those separating degenerate vacua which are adjacent in the field space. We denote these walls as adjacency walls. In the case that Z(N) terms are small compared with the U(1) terms, the SSB of U(1) generates strings first and then adjacency walls bounded by strings are generated after the SSB of Z(N). For symmetries larger than Z(3), nonadjacent vacua exist, and we regard walls separating them as nonadjacency walls. These walls are unstable if U(1) is a good approximation. If the discrete symmetry is broken via multiple steps, then we arrive at a complex structure where one kind of wall is wrapped by another type. On the other hand, if the symmetry is broken in different directions independently, then walls generated from the different breaking chains are blind to each other.