Length scales, collective modes, and type-1.5 regimes in three-band superconductors

The recent discovery of iron pnictide superconductors has resulted in a rapidly growing interest in multiband models with more than two bands. In this work we specifically focus on the properties of three-band Ginzburg-Landau models which do not have direct counterparts in more studied two-band models. First we derive normal modes and characteristic length scales in the conventional $U\left(1\right)$ three-band Ginzburg-Landau model as well as in its time-reversal symmetry-broken counterpart with $U\left(1\right)\times Z_2$ symmetry. We show that, in the latter case, the normal modes are mixed phase-density collective excitations. A possibility of the appearance of a massless mode associated with fluctuations of the phase difference is also discussed. Next we show that gradients of densities and phase differences can be inextricably intertwined in vortex excitations in three-band models. This can lead to very long-range attractive intervortex interactions and the appearance of type-1.5 regimes even when the intercomponent Josephson coupling is large. In some cases it also results in the formation of a domainlike structure in the form of a ring of suppressed density around a vortex across which one of the phases shifts by $\pi$. We also show that field-induced vortices can lead to a change of broken symmetry from $U\left(1\right)$ to $U\left(1\right)\times Z_2$ in the system. In the type-1.5 regime, it results in a semi-Meissner state where the system has a macroscopic phase separation in domains with broken $U\left(1\right)$ and $U\left(1\right)\times Z_2$ symmetries.

Figure 1

Ground-state phases of the three components as function of ${\eta }_{12}$ (here ${\varphi }_3=0$ fixes the gauge). The GL parameters are ${\alpha }_i=1$, ${\beta }_i=1$, and ${\eta }_{13}={\eta }_{23}=3$. For intermediate values of ${\eta }_{12}$ the ground state exhibits discrete degeneracy [symmetry is $U\left(1\right)\times Z_2$ rather than $U\left(1\right)$] since the energy is invariant under the sign change ${\varphi }_2\to -{\varphi }_2,{\varphi }_3\to -{\varphi }_3$. For large ${\eta }_{12}$ we get ${\varphi }_2-{\varphi }_3=\pi$, implying that $|{\psi }_3|=0$, and so there is a second transition from $U\left(1\right)\times Z_2$ to $U\left(1\right)$ and only two bands at the point (d). Here, the phases were computed in a system with only passive bands, though systems with active bands exhibit the same qualitative properties except for the transition to $U\left(1\right)$ and two bands only (i.e., active bands have nonzero density in the ground state).

Figure 2

Ground state, eigenspectrum, and eigenvectors of the mass matrix. The $x$ axis gives the two parameters ${\eta }_{13}={\eta }_{23}$ while the other parameters are ${\alpha }_1=-3$, ${\beta }_1=3$, ${\alpha }_2=-3$, ${\beta }_2=3$, ${\alpha }_3=2$, ${\beta }_3=0.5$, and ${\eta }_{12}=2.25$. The eigenvectors are sorted according to corresponding eigenvalue, starting with the largest. The smallest eigenvalue is the zero mode associated with the spontaneously broken $U\left(1\right)$ symmetry. At ${\eta }_{13}={\eta }_{23}\approx -3.69$ there is a transition from $U\left(1\right)$ to $U\left(1\right)\times Z_2$. Eigenvalue 5 (c) becomes zero at the transition point, so there appears a divergent length scale at this point which corresponds to the eigenvector in (h); i.e., the phase difference mode becomes a scaleless collective excitation. Observe that, in the $U\left(1\right)$ region, the eigenvectors exhibit no mixing between densities and phases, whereas in the $U\left(1\right)\times Z_2$ region there is in fact not a single eigenvector that is not a mixing of phases and densities. Then, perturbations of densities are generically associated with perturbations of the phase differences in this regime. Arrows in (g) and (h) illustrate the variation of the fields associated with the corresponding collective modes. Length of the arrows corresponds to modulus fluctuation and the direction of the arrow corresponds to phase fluctuation. Order parameter of component 1 in red, component 2 in blue and component 3 in black.

Figure 3

Vortex cluster exhibiting an internal $Z_2$ state in a frustrated three-band superconductor, showing (a) the magnetic field, and the densities of the different condensates are (b) $|{\psi }_1{|}^2$, (c) $|{\psi }_2{|}^2$, and (d) $|{\psi }_3{|}^2$. To monitor the relative phase differences, we use (e) $|{\psi }_1\left|\right|{\psi }_2|\sin \left({\varphi }_{12}\right)$, (f) $|{\psi }_1\left|\right|{\psi }_3|\sin \left({\varphi }_{13}\right)$, and (g) $|{\psi }_2\left|\right|{\psi }_3|\sin \left({\varphi }_{23}\right)$. (h) The energy density and the supercurrents in each condensate, (i) $J_1$, (j) $J_2$, and (k) $J_3$, are shown. The GL parameters used for this simulation are ${\alpha }_1=-3$, ${\beta }_1=3$, ${\alpha }_2=-3$, ${\beta }_2=3$, ${\alpha }_3=2$, ${\beta }_3=0.5$, ${\eta }_{12}=2.25$, ${\eta }_{13}=-3.7$, ${\eta }_{23}=-3.7$. Thus, they correspond to the $U\left(1\right)$ region in Fig. 2, but close to the transition point to $U\left(1\right)\times Z_2$ symmetry. In the ground state, all the phases are locked (${\overline{\varphi }}_1={\overline{\varphi }}_2={\overline{\varphi }}_3$) as a consequence of the Josephson couplings ${\eta }_{12}={\eta }_{13}=-3.7$ dominating the interaction. Inside the vortex cluster the third condensate is depleted, so the coupling terms ${\eta }_{i3}|{\psi }_i\left|\right|{\psi }_3|\cos \left({\varphi }_{i3}\right),\{i=1,2\}$, become much weaker while the term $|{\psi }_1\left|\right|{\psi }_2|{\eta }_{12}\cos \left({\varphi }_{12}\right)$ becomes dominant. In sufficiently dense vortex matter, the ground state is changed due to the dominating antilocking interaction between components 1 and 2. This results in a $U\left(1\right)\times Z_2$ state inside the vortex cluster, as can be seen from the phase-difference plots (e–g). (Note that in the very center of the vortex cluster this quantity is small because of small values of the prefactors $|{\psi }_i\left|\right|{\psi }_j|$.) A closer inspection of (b) and (c) reveals that vortex cores in both densities do not necessarily superimpose [which can also be seen from the supercurrents in (i) and (j)] and so they are fractional vortices. This fractionalization occurs at the boundary of the cluster, while the vortex in the middle is a composite one-quantum vortex.

Figure 4

Interacting vortex clusters with internal $Z_2$ symmetry in a frustrated superconductor. The figure represents a state where the energy is well minimized with respect to all variables except the relative positions of the weakly interacting well-separated clusters. The GL parameters and displayed quantities and panel labels are the same as in Fig. 3. The analysis of the eigenvalues in Fig. 2 shows that there is a mode with a very small mass, associated with the eigenvector $[0,0,0,1,-1,0]$. It corresponds to the mode associated with phase-difference fluctuations and it has the largest recovery length scale. This is indeed visible in (e–g). The phase difference ${\varphi }_{12}$ (e) recovers much more slowly than the magnetic field (a) and the condensate densities (b–d). Clusters clearly interact at a distance greatly exceeding the length scales of density modulation and the magnetic penetration depth, as this mode stretches out between them.

Figure 5

A vortex cluster surrounded by a $\pi$ wall. Here again displayed quantities and panel labels are the same as in Fig. 3. The GL parameters are ${\alpha }_1=-1$, ${\beta }_1=1$, ${\alpha }_2=1$, ${\beta }_2=0.5$, ${\alpha }_3=3$, ${\beta }_3=0.5$, ${\eta }_{12}=-2$, ${\eta }_{13}=2.7$, and ${\eta }_{23}=-4$. In the ground state, the phases are locked (${\overline{\varphi }}_1={\overline{\varphi }}_2={\overline{\varphi }}_3$), but frustration occurs as ${\eta }_{13}=2.7$ gives an antilocking interaction (i.e., the term $|{\psi }_1\left|\right|{\psi }_3|{\eta }_{13}\cos \left({\varphi }_{13}\right)$ is minimal for ${\varphi }_{13}=\pi$). As vortices are introduced in the system, the superfluid densities are depleted. It is clear from visual inspection that the vortices in the second band are larger than those of the first. Thus, the effective coupling ${\tilde{\eta }}_{23}$ decreases faster than ${\tilde{\eta }}_{13}$ and so inside the vortex cluster the preferred phase becomes ${\varphi }_1={\varphi }_2={\varphi }_3+\pi$. Since the third band has much smaller density than the other bands, the energetically cheapest way of coping with this is to create a domain-wall-like object where $|{\psi }_3|$ becomes very small. It does not cost much energy to have a large phase gradient density, so that ${\psi }_3$ quickly picks up a $\pi$ shift in its phase. As a result the density of $|{\psi }_3|$ is suppressed not only in the vortex cores but also in a ring surrounding the vortex cluster as can be clearly seen in (d).

Figure 6

A single vortex in a system exhibiting $\pi$-wall solutions. The interband coupling coefficients are ${\eta }_{12}=-2$, ${\eta }_{13}=2.7$, and ${\eta }_{23}=-4$. Displayed quantities here are (a) the magnetic flux and densities of each condensate (b) $|{\psi }_1{|}^2$, (c) $|{\psi }_2{|}^2$, and (d) $|{\psi }_3{|}^2$; (e) total energy density and supercurrents (f) $J_1$, (g) $J_2$, and (h) $J_3$; and (i) phase difference ${\varphi }_{13}$. The parameters of the system are identical to those given in Fig. 5. The $\pi$ wall can be seen from the double dip in density of the third band as can be seen in (j), as well as from the phase difference plotted in (i). Thus, ${\psi }_3$ is zero in the center, it recovers slightly, and then it drops again on a circular area at a certain distance from the vortex center. At the second drop, the phase ${\varphi }_3$ picks up an extra phase $\pi$, as can be seen from the plot of ${\varphi }_{13}$ in (i).