Ginzburg-Landau energy of multiband superconductors with interband pairing

We derive microscopically the Ginzburg-Landau free-energy functional for a superconductor in which the Cooper pairs are formed not only by quasiparticles from the same band, but also by quasiparticles from different bands. In the simplest case of an $s$- or $d$-wave pairing in a two-band system, the order parameter has three components describing two intraband and one interband pair condensates. The interband pairing-specific terms in the free energy bring about some qualitative features in the phase diagram, for example, time-reversal symmetry-breaking superconducting states are stabilized at low temperatures. We also show that, despite the fact that the gradient energy of the interband component turns negative at a sufficiently large band splitting, a nonuniform superconducting instability is generally suppressed due to the hybridization of the intraband and interband condensates.

Figure 1

The Fermi surfaces in the bands 1 and 2 located within the BCS pairing shell (the shaded annulus). In general, the Fermi surfaces can be anisotropic.

Figure 2

The functions $f_{1,2,3}\left(x\right)$ which determine the dependence of the GL free-energy coefficients on the band splitting ${\mathcal{E}}_b$ [see Eqs. (56) and (63)], with $x={\mathcal{E}}_b/4\pi T_c$. Note that $f_1\left(x\right)$ changes sign at $x\simeq 0.30$.

Figure 3

Stable states of the free energy (70) for $J>0$. The states I and II are TR invariant, while the states III and IV break TR symmetry. The black lines correspond to the continuous phase transitions between isolated stable minima. At the blue line, the minima of Eq. (70) are infinitely degenerate, as shown in Figs. 4 and 5.

Figure 4

The line of continuously degenerate minima of the free energy (70) at $q=p^2/4$ and $p<2$. The empty and filled circles show the states II and IV, respectively.

Figure 5

The line of continuously degenerate minima of the free energy (70) at $q=p^2/4$ and $p>2$. The empty and filled circles show the states II and III, respectively.

Figure 6

Stable states of the free energy (70) for $J<0$. The state I is TR invariant, while the state III breaks TR symmetry.