66
Topological defects
V(,)
..........
"........
..........
c'
----et>
-11
11
Figure 3.1. Effective potential for~4 theory.
In the following sections, we shall develop the theory of domain walls,
cosmic strings and magnetic monopoles in turn.
3.2 Domain walls
When the effective potential in a field theory has two degenerate minima with
V = 0, two-dimensional solutions of the field equations with finite energy per
unit area can occur. The simplest examples of such domain· wall solutions [2] are
obtained from the Lagrangian density
{, = !a"tI>8"tI> - V(tI»
(3.1)
with
V(tI» = ~(tI>2 _ ,,2)2
(3.2)
4
where tI> is a real scalar field and A and '7 are real constants. The potential V has
minima with V = 0 at tI> = ±" and a maximum at tI> = 0 (see figure 3.1). The
idea is to construct a static solution for which tI> evolves from one minimum for
z -+ -00 to the other minimum for z -+ +00. In that case, the domain wall is in
the x-y plane. Clearly, we can construct such solutions with the domain wall in
any chosen plane.
In general, for a static solution where tI> depends only on z, the field equation
is
d
2
t1> = V'(tI».
(3.3)
dz2