Three-band superconductivity and the order parameter that breaks time-reversal symmetry

We consider a model of multiband superconductivity, inspired by iron pnictides, in which three bands are connected via repulsive pair-scattering terms. Generically, three distinct superconducting states arise within such a model. Two of them are straightforward generalizations of the two-gap order parameter while the third one corresponds to a time-reversal symmetry-breaking order parameter, altogether absent within the two-band model. Potential observation of such a genuinely frustrated state would be a particularly vivid manifestation of the repulsive interband interactions being at the root of iron-based high-temperature superconductivity. We construct the phase diagram of this model and discuss its relevance to the iron pnictides family of high-temperature superconductors. We also study the case of the Josephson coupling between a two-band $s^{\prime }$ or $s\pm$ superconductor and a single-gap $s$-wave superconductor and the associated phase diagram.

Figure 1

(Color online) Plot of $\kappa$ (solid line) and $\text{cos}\phi$ (dashed line) for ${\tilde{\Delta }}_3$ at $T=0.95T_c$. This OP exists as a distinct solution only at a vicinity of the degenerate point $\lambda =\eta$. On the left it merges with ${\tilde{\Delta }}_1$ and on the right crosses ${\tilde{\Delta }}_2$ at $\text{cos}\phi =1$ and disappears. The interval is asymmetric with respect to the degenerate point.

Figure 2

(Color online) Comparison between the ${\mathcal{F}}_{\text{SC}}-{\mathcal{F}}_N$ for ${\tilde{\Delta }}_1$ (green), ${\tilde{\Delta }}_2$ (blue), and ${\tilde{\Delta }}_3$ (red). We show calculation on the right side of the $\lambda =\eta$ point for $T=0.95T_c$. The free energy of the real two-gap solution is constant and of the real three-gap one decreases monotonically. The complex OP minimizes the free energy in a small interval. The transition between ${\tilde{\Delta }}_3$ and ${\tilde{\Delta }}_2$ is discontinuous.

Figure 3

(Color online) Plot of ${\kappa }_0$ (solid line) and $\text{cos}{\phi }_0$ (dashed line) for ${\tilde{\Delta }}_3$ at $T=0$. This OP exists only for $\lambda ∊\left(0,{\lambda }_{c0}\right)$, where ${\lambda }_{c0}>\eta$.

Figure 4

(Color online) Comparison between the ${\mathcal{E}}_{\text{SC}}-{\mathcal{E}}_N$ for ${\tilde{\Delta }}_1$ (green), ${\tilde{\Delta }}_2$ (blue), and ${\tilde{\Delta }}_3$ (red). The first one is never a ground state for $\lambda \ne 0$. The energies for ${\tilde{\Delta }}_2$ and ${\tilde{\Delta }}_3$ merge for some ${\lambda }_{cr}>\eta$.

Figure 5

(Color online) Suggested phase diagram of the three-band model. There are three possible SC OPs. The line separating the TRSB and real three-gap OP is most likely first-order phase transition line.

Figure 6

Schematic representation of the TRSB order parameter in the case of three positive interband couplings (left) and two negative and a positive interband couplings (right). The frustration is resolved in a different but related way.

Figure 7

(Color online) Comparison of the $T_c$ for ${\tilde{\Delta }}_1$ (blue) and ${\tilde{\Delta }}_2$ (red) for $b=0$ (solid line), $b=0.5$ (dashed line), and $b=1$ (dotted line). $\eta$ is fixed. For small $\lambda$ the two-gap solution is the first to appear for $b<1$. At the crossing point there is degeneracy and complex $\tilde{\Delta }$ is possible.

Figure 8

(Color online) Comparison between the ${\mathcal{F}}_{\text{SC}}-{\mathcal{F}}_N$ for ${\tilde{\Delta }}_1$ (green), ${\tilde{\Delta }}_2$ (blue), and ${\tilde{\Delta }}_3$ (red) for $b=0.5$. We show calculation for $T=0.95T_c$ at the vicinity of the $T_{ci}$ crossing point $\lambda \approx 0.71\eta$. The interval for which the complex OP minimizes the free energy is smaller but still exists.

Figure 9

(Color online) Comparison between the ${\mathcal{E}}_{\text{SC}}-{\mathcal{E}}_N$ for the different order parameters. Here $b=0.5$. The energies for ${\tilde{\Delta }}_2$ and ${\tilde{\Delta }}_3$ cross for ${\lambda }_{cr}\approx 0.65$.