Extreme type-II superconductors in a magnetic field: A theory of critical fluctuations

A theory of critical fluctuations in extreme type-II superconductors subjected to a finite but weak external magnetic field is presented. It is shown that the standard Ginzburg-Landau representation of this problem can be recast, with help of a mapping, as a theory of a new ``superconductor,'' in an effective magnetic field whose overall value is zero, consisting of the original uniform field and a set of neutralizing unit fluxes attached to N_Φ fluctuating vortex lines. The long-distance behavior of this theory is governed by a phase transition line in the $(H,T)$ plane, T_Φ(H), along which the new ``superconducting'' order parameter Φ(r) attains long-range order. Physically, this phase transition arises through the proliferation, or ``expansion,'' of thermally generated infinite vortex loops in the background of field-induced vortex lines. Simultaneously, the field-induced vortex lines lose their effective line tension relative to the field direction. It is suggested that the critical behavior at T_Φ(H) belongs to the universality class of the anisotropic Higgs-Abelian gauge theory, with the original magnetic field playing the role of ``charge'' in this fictitious ``electrodynamics'' and with the absence of reflection symmetry along H giving rise to dangerously irrelevant terms. At zero field, Φ(r) and the familiar superconducting order parameter Ψ(r) are equivalent, and the effective line tension of large loops and the helicity modulus vanish simultaneously, at T=Tc0. In a finite field, however, these two forms of ``superconducting'' order are not the same and the ``superconducting'' transition is generally split into two branches: the helicity modulus typically vanishes at the vortex lattice melting line Tₘ(H), while the line tension and associated {Φ} order disappear only at T_Φ(H). We expect T_Φ(H)>Tₘ(H) at lower fields and T_Φ(H)=Tₘ(H) for higher fields. Both {Φ} and Ψ order are present in the Abrikosov vortex lattice [T<Tₘ(H)] while both are absent in the true normal state [T>T_Φ(H)]. The intermediate {Φ}-ordered phase, between Tₘ(H) and T_Φ(H), contains precisely N_Φ field-induced vortices having a finite line tension relative to H and could be viewed as a ``line liquid'' in the long-wavelength limit. The consequences of this ``gauge theory'' scenario for the critical behavior in high-temperature and other extreme type-II superconductors are explored in detail, with particular emphasis on the questions of three-dimensional XY versus Landau level scaling, physical nature of the vortex ``line liquid'' and the true normal state (or vortex ``gas''), and fluctuation thermodynamics and transport. It is suggested that the empirically established ``decoupling transition'' may be associated with the loss of integrity of field-induced vortex lines as their effective line tension disappears at T_Φ(H). A ``minimal'' set of requirements for the theory of vortex lattice melting in the critical region is also proposed and discussed. The mean-field-based description of the melting transition, containing only field-induced London vortices, is shown to be in violation of such requirements.