Ginzburg-Landau theory of type II superconductors in magnetic field

Thermodynamics of type II superconductors in electromagnetic field based on the Ginzburg-Landau theory is presented. The Abrikosov flux lattice solution is derived using an expansion in a parameter characterizing the “distance” to the superconductor-normal phase transition line. The expansion allows a systematic improvement of the solution. The phase diagram of the vortex matter in magnetic field is determined in detail. In the presence of significant thermal fluctuations on the mesoscopic scale (for example, in high $T_c$ materials) the vortex crystal melts into a vortex liquid. A quantitative theory of thermal fluctuations using the lowest Landau level approximation is given. It allows one to determine the melting line and discontinuities at melt, as well as important characteristics of the vortex liquid state. In the presence of quenched disorder (pinning) the vortex matter acquires certain “glassy” properties. The irreversibility line and static properties of the vortex glass state are studied using the “replica” method. Most of the analytical methods are introduced and presented in some detail. Various quantitative and qualitative features are compared to experiments in type II superconductors, although the use of a rather universal Ginzburg-Landau theory is not restricted to superconductivity and can be applied with certain adjustments to other physical systems, for example, rotating Bose-Einstein condensate.

Figure 1

(Color online) Schematic magnetic phase diagram of a type II superconductor.

Figure 2

(Color) Magnetic phase diagram of high $T_c$. (a) Experimentally determined phase diagram of LaSCO. From 43. (b) Theoretical phase diagram advocated in this article.

Figure 3

(Color online) Symmetry of the vortex lattice. Unit cell.

Figure 4

(Color) Superflow around the vortex centers in the hexagonal lattice.

Figure 5

(Color) Convergence of the bifurcation perturbation theory.

Figure 6

Feynman rules for vortex lattice.

Figure 7

Two loops connected diagrams contributing to free energy.

Figure 8

Feynman rules and the diagrams in the homogeneous phase: (a) propagator, (b) vertex, (c) the two loop energy correction.

Figure 9

(Color online) Free energy in liquid. The curve $T0$ is the Gaussian approximation, while $T1$,… are higher order renormalized perturbation theory results. Optimized perturbation theory gives curves 1,2,… and finally BP lines are the Borel-Pade results.

Figure 10

Additional diagram of the renormalized perturbation theory shown in (a). Bubbles or cacti diagrams summed by the optimized expansion are shown in (b)–(d). A diagram which is not of that type is shown in (f).

Figure 11

(Color online) The melting point and the spinodal point of the crystal. The free energies of the crystalline and the liquid states are equal at melt, while metastable crystal becomes unstable at spinodal point.

Figure 12

(Color) Phase diagram for YBCO.

Figure 13

Universal function $d\left(a_T\right)$ determining the shift of the melting line due to disorder.

Figure 14

(Color online) $\mathrm{Nb}{\mathrm{Se}}_2$ phase diagram.

Figure 15

(Color online) Melting line and order-disorder lines in layered superconductor BSCCO (10).