Chiral $\mathbb{C}{P}^2$ skyrmions in three-band superconductors

It is shown that under certain conditions, three-component superconductors (and, in particular, three-band systems) allow stable topological defects different from vortices. We demonstrate the existence of these excitations, characterized by a $\mathbb{C}{P}^2$ topological invariant, in models for three-component superconductors with broken time-reversal symmetry. We term these topological defects “chiral $G{L}^{\left(3\right)}$ skyrmions,” where “chiral” refers to the fact that due to broken time-reversal symmetry, these defects come in inequivalent left- and right-handed versions. In certain cases, these objects are energetically cheaper than vortices and should be induced by an applied magnetic field. In other situations, these skyrmions are metastable states, which can be produced by a quench. Observation of these defects can signal broken time-reversal symmetry in three-band superconductors or in Josephson-coupled bilayers of $s_{\pm }$ and $s$-wave superconductors.

Figure 1

Example of unusual observable magnetic field configuration of chiral skyrmions.

Figure 2

Representation of the vacuum submanifold (top) for $({\alpha }_a,{\beta }_a)=(-1,1)$ and ${\eta }_{ab}=-3$. The image shows the potential energy as a function of the phase differences: ${\phi }_2$ and ${\phi }_3$, minimized with respect to all moduli degrees of freedom [while ${\phi }_1$ is set to zero by $\text{U(1)}$ invariance associated with simultaneous change of all phases]. Red and green dots show inequivalent ${\mathbb{Z}}_2$ ground states, and the lines connecting them represent four different kinds of domain-wall trajectory over the field manifold. Black dots are ground states located farther than $2\pi$ in the phase differences. The second line gives a schematic representation of various ${\mathbb{Z}}_2$ domain walls in three-band superconductors with different frustrations of phase angles, shown by arrows of different colors. The pink line schematically shows the phase difference between red and green arrow, interpolating between the two inequivalent ground states.

Figure 3

Single-charge chiral skyrmion for three mirror passive bands $({\alpha }_a,{\beta }_a)=(1,1)$ and Josephson coupling constants ${\eta }_{ab}=-3$. Here, ${\gamma }_{ab}=0.8$ and the gauge coupling constant is $e=0.6$. Displayed quantities are the magnetic flux (A) and the sine of phase differences $\sin \left({\phi }_{12}\right)$ (B), $\sin \left({\phi }_{13}\right)$ (C). Condensate densities $|{\psi }_1^2|$, (D), $|{\psi }_2^2|$, (E) and $|{\psi }_3^2|$, (F) are represented on the second line. The corresponding supercurrent densities $|J_1|$, (G), $|J_2|$, (H) and $|J_3|$, (I) are displayed on the third line. To avoid redundant informations, the total energy density is not displayed. It qualitatively follows the magnetic flux shown in panel (A).

Figure 4

Skyrmion with $\mathcal{Q}=6$ topological charge (which implies that it carries six flux quanta and consists of 18 fractional vortices). Displayed quantities are the magnetic flux (A) and the sine of phase differences $\sin \left({\phi }_{12}\right)$ (B), $\sin \left({\phi }_{13}\right)$ (C). Condensate densities $|{\psi }_1^2|$ (D), $|{\psi }_2^2|$ (E), and $|{\psi }_3^2|$ (F) are represented on the second line. The corresponding supercurrent densities $|J_1|$ (G), $|J_2|$ (H), and $|J_3|$ (I) are displayed on the third line. Parameters are the same as in Fig. 3.

Figure 5

$\mathcal{Q}=2$ quantum soliton in a system with two identical passive bands $({\alpha }_a,{\beta }_a)=(1,1)$ ($a=1,2$) coupled to a third active band with substantial disparity in the ground-state densities $({\alpha }_3,{\beta }_3)=(-2.75,1)$. Josephson coupling constants are ${\eta }_{12}={\eta }_{13}={\eta }_{23}=-3$. The system is in a strongly type-II regime $e=0.08$, the solutions here are stable even in the absence of biquadratic density interaction, i.e., ${\gamma }_{ab}=0$. Displayed quantities are the magnetic flux (A) and the sine of phase differences $\sin \left({\phi }_{12}\right)$ (B) $\sin \left({\phi }_{13}\right)$ (C). Condensate densities $|{\psi }_1^2|$ (D), $|{\psi }_2^2|$ (E), and $|{\psi }_3^2|$ (F) are represented on the second line. The corresponding supercurrent densities $|J_1|$ (G), $|J_2|$ (H), and $|J_3|$ (I) are displayed on the third line.

Figure 6

$\mathcal{Q}=5$ quantum soliton in a system with two identical passive bands as in Fig. 5 coupled to a third active band with disparity in the ground-state densities $({\alpha }_3,{\beta }_3)=(-1.5,1)$. Josephson coupling constants are ${\eta }_{23}=-3$ and ${\eta }_{12}={\eta }_{13}=1$. Here, $e=0.2$ and there is no density-density interaction term ${\gamma }_{ab}=0$. The system is shaped as a pentagon deformed by the vortices of the strong active band carrying larger fractions of flux quantum. Displayed quantities are the same as in the previous pictures, e.g., Fig. 5.

Figure 7

$\mathcal{Q}=5$ quantum soliton in a system with within the same parameter set as in Fig. 6 apart from $({\alpha }_3,{\beta }_3)=(-0.5,1)$. Displayed quantities are the same as in the previous pictures, e.g., Fig. 5.

Figure 8

$\mathcal{Q}=11$ quantum multisoliton in a system with three identical passive bands as in Fig. 13. The current soliton is not made out of one but two stabilized domain walls thus being a homogeneous bi-ring configuration.(B) and (C) clearly display the alternating different ground states. Since the three bands are identical, the magnetic field rather homogeneously spreads all along the solitons. Displayed quantities are the same as in the previous pictures, e.g., Fig. 5. Note that while going counterclockwise along the outer ring, the fractional vortices have order band-“1,2,3”. For the inner ring they are ordered as band-“1,3,2”. The origin of this is discussed in Sec.  4b.

Figure 9

Triring chiral skyrmion. The configuration carries total charge $\mathcal{Q}=36$ in a system with two identical passive bands $({\alpha }_1,{\beta }_1)=({\alpha }_2,{\beta }_2)=(1,1)$ coupled to a third active band with $({\alpha }_3,{\beta }_3)=(-1,1)$. Josephson coupling constants are ${\eta }_{23}=-3$ and ${\eta }_{12}={\eta }_{13}=1$. $e=0.7$. Panels are the same as usual, e.g., Fig. 5.

Figure 10

Energies per flux quantum of the skyrmions carrying $N$ flux quanta. The energy is given in units of the energy of the energetically cheapest (either vortex or skyrmion) single quantum excitation (A). (B) shows $\frac{E\left(N\right)-E(N-1)}{E(N=1)}$ as a function of the number of flux quanta. When this quantity is less than one, it is energetically preferred to have an $N$-quantum skyrmion than having a $(N-1)$ skyrmion plus one isolated vortex. The criterion for thermodynamical stability $\frac{H_{c1}^2}{2}-\left|F_{\mathrm{GS}}\right|$, where the condensation energy is $F_{\mathrm{GS}}\equiv F(\langle {\psi }_a\rangle ,0)$, is shown on (C). Here, the dependence of the solutions on $\gamma$ and $N$ is investigated, while the gauge coupling is fixed at $e=0.3$. Other parameters are $({\alpha }_a,{\beta }_a)=(1,1)$ and ${\eta }_{ab}=-3$. Colors and symbols associated to different values of ${\gamma }_{ab}$ (shown on the picture) are the same over three panels. Note that the reason why curves with high ${\gamma }_{ab}$ terminate for smaller $N$ is that the size of the skyrmion becomes comparable to the size of the numerical domain. To avoid any finite size effect, we chose to skip the corresponding points.

Figure 11

Energies per flux quantum of the chiral skyrmions in the units of the energy of the energetically cheapest (either vortex or skyrmion) single quantum excitation (A). Curves with same color and symbols on different panels have same parameters. (B) shows that it is always beneficial (within a parameter range) to have a higher-charge skyrmion than a lower charge one plus an isolated one quantum vortex. The criterion for thermodynamical stability of $N$-quantum solitons $\frac{H_{c1}^2}{2}-\left|{\mathcal{F}}_{\mathrm{GS}}\right|$, where ${\mathcal{F}}_{\mathrm{GS}}\equiv \mathcal{F}(\langle {\psi }_a\rangle ,0)$ is the condensation energy (C). The dependence of the solutions on $e$ and $N$ is investigated, for a strength of the density-density interactions ${\gamma }_{ab}=0.8$. Other parameters are the same as in Fig. 10. Here again, curves are truncated when the soliton's size becomes comparable to the numerical domain.

Figure 12

Relaxation of a randomly perturbed chiral skyrmion. Displayed quantities are the energy density, $\mathrm{Im}\left({\psi }_1^{*}{\psi }_2\right)$ and $|{\psi }_1{|}^2$. The parameters are the same as in Fig. 3, but vanishing biquadratic density interactions ${\gamma }_{ab}=0$ and $e=0.3$. Thus it is only metastable. The snapshots show the state of the system at different stages of the energy minimization algorithm after the applied perturbation. (Top) $\mathcal{Q}=6$ chiral skyrmion with initial white noise of 70$\%$ of the ground-state values. The configurations relaxes to a chiral skyrmion. On the bottom panel, a perturbation of a metastable charge $\mathcal{Q}=3$ soliton with an initial noise $P=0.8$. Here, the noise is strong enough to break up the domain wall. The soliton thus relaxes to ordinary type-II vortices (one can clearly see the disappearance of the domain wall between blue and red area in the middle row). The last snapshot in the lower configuration does not represent a stationary configuration: the vortices repel each other and are in process of drifting apart.

Figure 13

$\mathcal{Q}=9$ quantum configuration of mixed vortices and skyrmions in a system with three identical passive bands as in Fig. 12. This configuration is made out of a skyrmion surrounding two ordinary vortices. It is known, from energy considerations, that the interaction is short-range attractive. The interaction with vortices deforms the skyrmions. This shows that it is long-range repulsive.

Figure 14

(a) Schematic picture of how soliton interactions are computed. This figure shows the interaction between two single quanta solitons, each consisting of three fractional vortices shown in green, blue, and red. This generalizes easily to larger solitons. One soliton ($\mathrm{x}$) is placed in the origin, while the second ($\mathrm{y}$) is placed at a distance $R$ at an angle $v$. Consequently, as $v$ is varied, the relative orientation of the solitons changes. Case 1 shows a system of two solitons with identical chiralities (same ordering $o$), while case 2 shows two solitons with opposite chiralities (different $o$), although the mirrored soliton is not necessarily stable. A schematic comparison of solitons with different ordering $o$ is displayed in (b). In case 2, the gradients in phase difference due to the fractional vortices naturally interpolate between two ${\mathbb{Z}}_2$ states. For this reason, case 2 is energetically preferable over case 1, and it was verified numerically. Finally, (c) gives a schematic view of the merging of two single quanta solitons. In order to merge, they should have same ordering but opposite orientation.

Figure 15

(a) Interaction energy of two single quantum skyrmions. One soliton is placed at the origin, the interaction energy is plotted as a function of the position and relative orientation of the second soliton. The interaction energy is maximal when $v=0$, while it is minimal for the opposite orientation, $v=\pi$. The strength of the interaction decreases with the separation $R$. The model parameters are the same as in Fig. 3. (b) Interaction energy of two $\mathcal{Q}=2$ quanta solitons, for the same parameters as in Fig. 5. Note that the skyrmion has two-fold symmetry (it is invariant under global rotations of $\pi$). The minimum energy is found for the relative orientation $v\pm \pi /2$.

Figure 16

$\mathcal{Q}=1$ soliton for the $\text{U}\left(3\right)$ symmetric model broken by Josephson interactions of the form (5.42), with $\Lambda =20$ and ${\eta }_0=1$. The quantities ${|1-|\psi |}^2|$ (A), and $|e_3^{†}\Psi |$ (B), measure the deviation from the $\sigma$ model. They converge to zero as $\Lambda$ is increased. (C) Energy density of the skyrmion. (D) Texture of the field $\mathbf{n}$, which is similar to that of a baby skyrmion.

Figure 17

Results of energy minimization with charge $\mathcal{Q}=2$ for $\Lambda =20$, and $e=1.0$ and varying ${\eta }_0$. First row shows the energy density while the second and third row displays the corresponding texture field. (A) ${\eta }_0=0.1$, (B) ${\eta }_0=0.2$, and (C) ${\eta }_0=0.3$) are for ${\eta }_0<{\eta }_0^{*}$ where interaction between skyrmions is attractive. Two $\mathcal{Q}=1$ skyrmions coalesce into either one $\mathcal{Q}=2$ skyrmion (A)–(C). Configuration displayed in (C) resemble bound state of two $\mathcal{Q}=1$ skyrmions. (D), ${\eta }_0=0.8$, has ${\eta }_0>{\eta }_0^{*}$, then in the repulsive channel. Here, the two $\mathcal{Q}=1$ skyrmions are repelling each other. So the snapshot in (D) shows a late but unconverged iteration (i.e., it represents a fairly converged pair of individual skyrmions that are, however, still drifting apart).

Figure 18

Interaction energy of two single quantum solitons. The the GL parameters are ${\alpha }_a=-20$, ${\beta }_a=20$, ${\gamma }_{ab}=20$, ${\eta }_{ab}=-1$, and $e=1$. Thus the potential part of the free energy density can be written as $U=\lambda (1-|{\psi }_1{|}^2-|{\psi }_2{|}^2-|{\psi }_3{|}^2{)}^2-{\eta }_{ab}|{\psi }_a\left|\right|{\psi }_b|\cos ({\phi }_a-{\phi }_b)$ with $\lambda =10$, i.e., the Hamiltonian features $\text{SU}\left(3\right)$ symmetry broken by Josephson interaction term.

Figure 19

$\mathcal{Q}=3$ quanta soliton in a system with three identical passive bands as in Fig. 3, except that there are no density-density interactions ${\gamma }_{ab}=0$ and $e=0.3$. Since the three bands are identical, the soliton makes a homogeneous ringlike configuration. Displayed quantities are the same as in rest of the paper.

Figure 20

$\mathcal{Q}=4$ quanta soliton in a system with two identical passive bands as in Fig. 5 coupled to a third active band with disparity in the ground-state densities $({\alpha }_3,{\beta }_3)=(-1.5,1)$. Josephson coupling constants are ${\eta }_{23}=-3$ and ${\eta }_{12}={\eta }_{13}=1$. $e=0.2$ and ${\gamma }_{ab}=0$.

Figure 21

$\mathcal{Q}=7$ quanta soliton in a system with two identical passive bands as in Fig. 5 coupled to a third active band with disparity in the ground-state densities $({\alpha }_3,{\beta }_3)=(-1,1)$. Josephson coupling constants are ${\eta }_{23}=-3$ and ${\eta }_{12}={\eta }_{13}=1$. $e=0.3$ and ${\gamma }_{ab}=0$.