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: MN if there exists M(t)( ) continuous both in and in 0 <= t <= 1, with M(0)( ) M and M(1)( ) N.

Figure 4 reproduced.


Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).