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 ${N}_{\ensuremath{\Phi}}$ fluctuating vortex lines. The long-distance behavior of this theory is governed by a phase transition line in the $(H,T)$ plane, ${T}_{\ensuremath{\Phi}}(H),$ along which the new ``superconducting'' order parameter $\ensuremath{\Phi}(\mathbf{r})$ 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 ${T}_{\ensuremath{\Phi}}(H)$ 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, $\ensuremath{\Phi}(\mathbf{r})$ and the familiar superconducting order parameter $\ensuremath{\Psi}(\mathbf{r})$ are equivalent, and the effective line tension of large loops and the helicity modulus vanish simultaneously, at ${T=T}_{c0}.$ 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 ${T}_{m}(H),$ while the line tension and associated \ensuremath{\Phi} order disappear only at ${T}_{\ensuremath{\Phi}}(H).$ We expect ${T}_{\ensuremath{\Phi}}(H)g{T}_{m}(H)$ at lower fields and ${T}_{\ensuremath{\Phi}}{(H)=T}_{m}(H)$ for higher fields. Both \ensuremath{\Phi} and $\ensuremath{\Psi}$ order are present in the Abrikosov vortex lattice $[Tl{T}_{m}(H)]$ while both are absent in the true normal state $[Tg{T}_{\ensuremath{\Phi}}(H)].$ The intermediate \ensuremath{\Phi}-ordered phase, between ${T}_{m}(H)$ and ${T}_{\ensuremath{\Phi}}(H),$ contains precisely ${N}_{\ensuremath{\Phi}}$ field-induced vortices having a finite line tension relative to $\mathbf{H}$ 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 $\mathrm{XY}$ 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 ${T}_{\ensuremath{\Phi}}(H).$ 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.