The Higgs field has a non-trivial self-interaction, which leads
to spontaneous symmetry breaking: in the lowest energy state the
symmetry of the potential (which includes the gauge symmetry) is
broken by the condensate.
In the simplest example, the spontaneously broken field is
described by a scalar field theory. In physics, one way of seeing
spontaneous symmetry breaking is through the use of Lagrangians.
Lagrangians, which essentially dictate how a system will behave,
can be defined in potential terms:
L = du(f)du(f) - V(f).
It is in this potential term V(f) that the action of symmetry
breaking occurs. The potential graph of this function has the look
and appearance of a Mexican hat.