Topological Solitons in Three-Band Superconductors with Broken Time Reversal Symmetry

We show that three-band superconductors with broken time reversal symmetry allow magnetic flux-carrying stable topological solitons. They can be induced by fluctuations or quenching the system through a phase transition. It can provide an experimental signature of the time reversal symmetry breakdown.

Figure 1

Schematic representation of various $Z_2$ domain walls in three-band superconductors with different frustrations of phase angles, shown by arrows of different colors. Pink line schematically shows phase difference between red and green arrow, interpolating between the two inequivalent ground states.

Figure 2

$N=2$ topological solitons for two similar passive bands $({\alpha }_i,{\beta }_i)=(1,1)$ with interband coupling ${\eta }_{12}=-3$. These bands have Josephson coupling ${\eta }_{13}={\eta }_{23}=1$ to the third band, which is active $({\alpha }_3,{\beta }_3)=(-2.5,1)$. The system is type-II with $e=0.07$ [we use coupling constant $e$ in (1) to parametrize inverse penetration length]. The panel $\mathbf{A}$ displays the magnetic field $B$. Panels $\mathbf{B}$ and $\mathbf{C}$ respectively display $({\psi }_1^{*}{\psi }_2-{\psi }_1{\psi }_2^{*})/2i$ and $({\psi }_1^{*}{\psi }_3-{\psi }_1{\psi }_3^{*})/2i$, showing the phase difference between two condensates. Second line, shows the densities of the different condensates $|{\psi }_1{|}^2$ ($\mathbf{D}$), $|{\psi }_2{|}^2$ ($\mathbf{E}$), $|{\psi }_3{|}^2$ ($\mathbf{F}$). The third line displays the supercurrent densities associated with each condensate $|J_1|$ ($\mathbf{G}$), $|J_2|$ ($\mathbf{H}$), $|J_3|$ ($\mathbf{I}$). Phase differences on panels $\mathbf{B}$ and $\mathbf{C}$ show that there is a closed domain wall since there are two areas with different phase lockings (blue and red) associated with two possible ground states. The solution consists of $N=2$ vortices which are fractionalized: indeed, the panels $\mathbf{D}$, $\mathbf{E}$, and $\mathbf{F}$ show separated highly asymmetric pairs of singularities of different condensates. Note the very complicated geometry of supercurrent densities shown on panels $\mathbf{G}$, $\mathbf{H}$, and $\mathbf{I}$.

Figure 3

$N=8$ quanta soliton for the same parameter set as in Fig. 2 except that $e=0.3$ and $({\alpha }_3,{\beta }_3)=(-1.5,1)$, giving less disparity in the ground state densities (displayed quantities are the same as in Fig. 2). The cores of vortices in each bands do not coincide. Note the complicated structure of currents in each band.

Figure 4

$N=5$ quanta soliton with $e=0.3$. With three identical passive bands $({\alpha }_i,{\beta }_i)=(1,1)$, with superconductivity induced by repulsion ${\eta }_{ij}=-3$ between the three condensates. Displayed quantities are the same as in Fig. 2.

Figure 5

$N=4$ quanta soliton for two similar passive bands coupled to a third active band. The parameter set used here is the same as in Fig. 3 except $({\alpha }_3,{\beta }_3)=(-0.5,1)$ and $e=0.2$. Displayed quantities are the same as in Fig. 2.

Figure 6

Energies of the solitons per flux quanta, in the units of the energy of a single ordinary vortex (left). When the electric charge increases (i.e., the penetration length decreases) solitons with smaller $N$ become unstable. The right panel shows cross sections of the magnetic field for solitons with $N\in$ [2, 9] (double-peak curves). The central curve corresponds to a cross section of a regular $N=1$ vortex. The parameters of the Ginzburg-Landau model used here are the same as in Fig. 4, which gives nearly axially symmetric magnetic field.

Figure 7

The soliton formation during energy relaxation of an initial state of expanding group of vortices in a circular system with open boundary conditions. First line displays the energy density. Second line shows the phase difference between condensates $({\psi }_1^{*}{\psi }_2-{\psi }_1{\psi }_2^{*})/2i$. When domain walls form they separate two inequivalent ground states (blue and red). Third line is the density of the first condensate $|{\psi }_1{|}^2$. Initial configuration has a high density of 13 vortices in the center. Repulsive type-II interaction makes all vortices move away from each other and escape the sample. In the process of energy minimization domain walls and $G{L}^{(3)}$ solitons form. Domain wall connected to boundaries quickly disappear. The final picture shows the resulting long-living state of a well-separated $N=4$ $G{L}^{(3)}$ soliton and a vortex. Parameter set used here is the same as in Fig. 4, with $e=0.4$.