Spontaneous ordering against an external field in nonequilibrium systems

We study the collective behavior of nonequilibrium systems subject to an
external field with a
dynamics characterized by the existence of non-interacting states.
Aiming at exploring the generality of the results, we consider
two types of models according to the nature of their state variables: (i)
a vector model, where interactions are proportional to the overlap
between the states, and
(ii) a scalar model, where interaction depends on the distance between
states. In both cases the system displays three phases:
two ordered phases, one parallel to the
field, and another orthogonal to the field; and a disordered phase.
The phase space is numerically characterized for each model in a fully
connected network.
By placing the particles on a small-world network, we show that, while a
regular lattice favors
the alignment with the field, the presence of long-range interactions
promotes the formation of the ordered
phase orthogonal to the field.