The role of the Manhattan distance in antiferromagnetic ordering

The lowest state of one- and two-dimensional antiferromagnetic Heisenberg systems for a given number of "up" and "down" spins shows preference for certain regular patterns of the minority spin direction. For the one-dimensional system this pattern is an even distribution with equal distances for the minority spins, whereas in the two-dimensional systems the Manhattan distance, instead of the Euclidean distance, determines the optimal configuration