Domain Walls and Their Experimental Signatures in $s+is$ Superconductors

Arguments were recently advanced that hole-doped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ exhibits the $s+is$ state at certain doping. Spontaneous breaking of time-reversal symmetry in the $s+is$ state dictates that it possess domain wall excitations. Here, we discuss what are the experimentally detectable signatures of domain walls in the $s+is$ state. We find that in this state the domain walls can have a dipolelike magnetic signature (in contrast to the uniform magnetic signature of domain walls $p+ip$ superconductors). We propose experiments where quench-induced domain walls can be stabilized by geometric barriers and observed via their magnetic signature or their influence on the magnetization process, thereby providing an experimental tool to confirm the $s+is$ state.

Figure 1

This figure shows the symmetry breaking pattern for a frustrated three-band superconductor. Surfaces show the potential energy as a function of the phase differences, at different temperatures. The blue line shows the ground state. Above $T_{{\mathbb{Z}}_2}$, phases are locked and the ground state is unique up to overall U(1) transformations. Below $T_{{\mathbb{Z}}_2}$ the ground state is degenerate and time-reversal symmetry is broken.

Figure 2

A geometrically stabilized domain wall in a nonconvex domain, at $T/T_c=0.8$ for the parameter set I. The domain wall is geometrically trapped, since to escape it should increase its length, which is energetically costly. The phase difference ${\phi }_{12}$ shows that during the cooling, domain walls were created and one has been stabilized by the sample’s geometry. The unfavorable phase differences at the domain wall affect the densities of the condensates. $|{\psi }_1{|}^2$ overshoots at the domain wall, while $|{\psi }_2{|}^2$ and $|{\psi }_3{|}^2$ are depleted. Note that the domain wall has a magnetic signature: spots of the dipolelike magnetic field, where the domain wall touches the bumps. It originates in features of the interband counterflow at the domain wall, discussed in the text. The upper right panel shows the contribution to the magnetic field of the second term in Eq. (2).

Figure 3

A domain wall stabilized by randomly located pinning centers. When cooled past $T_{{\mathbb{Z}}_2}$, domain walls are formed at random positions. Then, during the relaxation process, the quench-induced domain wall is stabilized against collapse by nearby pinning centers. Displayed quantities and physical parameters are the same as in Fig. 2. Here again, the domain wall has a dipolelike signature of the magnetic field where it is attached to the pinning centers.

Figure 4

Field cooled experiment for the parameter set II, under an applied field $H/{\mathrm{\Phi }}_{0}S=70$. First, the system is a two band (${\psi }_2=0$) and, thus, it is TRS. When cooled through $T_{{\mathbb{Z}}_2}$ (${\psi }_2\ne 0$) the system enters the BTRS regime and different regions pick up different ground state phase locking. The resulting DWs are stabilized against complete contraction by the already existing vortices.

Figure 5

Magnetization process of a three-band BTRS, when the zero field configuration has the geometrically stabilized domain wall (Fig. 2). First, vortex entry, way below $H_{c1}$, shown in the top row is a fractional vortex in ${\psi }_3$. This can be seen from the phase difference ${\phi }_{13}$ which winds $2\pi$. The red curve is the corresponding magnetization curve, while the blue curve is a reference magnetization, starting from a uniform ground state.