A loop is a continuous mapping from the circle into
the order parameter space:
->
M(
),
0 <=
<=
2
When we encircle the defect with a loop, we get a
loop in order parameter space as shown in figure 4 (reproduced below):
->
x(
)
is the loop in real space, and
->
M(x(
))
is the loop in order parameter space. Two loops
are equivalent if there is a continuous one-parameter family of loops
connecting one to the other:
M
N if there exists
M(t)(
)
continuous both in
and in 0 <= t <= 1, with
M(0)(
)
M
and
M(1)(
)
N.
Figure 4 reproduced.
Back
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).