In the three-dimensional vortex lattice, vacancies and interstitials are actually line defects. This gives them much greater significance than the usual point defects. As line defects, their chemical potential must be either zero (i.e. they are present in equilibrium) or infinite (i.e. a finite energy per unit length). We therefore expect two distinct types of phases, in which these defects are either bound or unbound.
To define an order parameter to distinguish these phases, let us work in the grand canonical ensemble. Imagine defining ``operators'' and , which remove or insert, respectively, one end of a vortex line at position . Consider the correlation function
This measures the free energy of an interstitial line present for a length z in the field direction. If interstitials are unbound, this free energy is zero, and will not decay for large z. On the other hand, if interstitials are bound, we expect exponential decay, where the inverse correlation length gives the free energy per unit length of an interstitial line. That is,
where is the defect chemical potential per unit length, and we have defined the constant asymptote of as the square of the expectation value . One may argue that the restriction to equal x and y in the definition of is inessential.
The field b arises in the so-called ``Boson mapping'' for assemblies of vortex lines. From work by M.P.A. Fisher and D.-H. Lee, it is known that b is in fact dual to the original Ginzburg-Landau order parameter . The existence of the complex order parameter b is thus related to the U(1) symmetry of the original Ginzburg-Landau theory. We will see that its existence allows for the possibility of additional types of glassy phases.