Microscopic properties of fractional vortices and domain walls in three-band $s+is$ superconductors

TL;DR

This work addresses the existence and properties of fractional vortices and domain walls in three-band $s+is$ superconductors by solving a fully self-consistent microscopic Bogoliubov-de Gennes model with interband Josephson coupling. The authors demonstrate stable fractional vortices carrying a fraction of the magnetic flux (e.g., $rac{ ext{$oxed{rac{1}{3}}$} imes ext{$ ext{flux quantum}$}}$ in symmetric cases) and domain-wall configurations, whose energy and magnetic-field distributions depend on temperature and band asymmetries. They compute tunneling conductance and core-state signatures, showing distinct STM fingerprints that differentiate fractional from conventional vortices, and reveal how domain walls modify the vortex structure and potential mobility. The results connect to experimental observations in multiband materials and highlight implications for anyon-like statistics and fluxonics in $s+is$ superconductors, providing concrete predictions for STM and magnetic-field imaging experiments.

Abstract

Several experimental observations of objects carrying fractional flux quanta in superconductors were recently reported. Here, we provide microscopic solutions for vortices carrying a variable fraction of magnetic flux quantum and domain walls in a three-band $s + is$ superconductor and investigate their properties. We obtain solutions in a fully self-consistent treatment of a microscopic three-band Bogoliubov-de-Gennes model. This demonstrates the characteristic patterns for the magnetic field distribution. The microscopic formalism allows for calculating tunneling conductance that may be used to distinguish fractional vortices from conventional single flux quanta vortices in Scanning Tunneling Microscopy.

Figures (8)

• Figure 1: Fractional vortex with phase winding only in the first component in a $U(1)^3$ three-component model. (a) Total current $J$ (Eq. (\ref{['eq:curr']})). (b-d) partial currents $J_1$, $J_2$, and $J_3$ (Eq. (\ref{['eq:par_curr']})). (e-g) absolute order parameter values $\Delta_1$, $\Delta_2$, and $\Delta_3$. (h) and (i) relative order parameter phases $\varphi_1-\varphi_2$ and $\varphi_1-\varphi_3$. The total magnetic flux in the system is $\Phi_0/3$. Square sample with linear size 32 with intraband coupling $V_{11}=V_{22}=V_{33} = 2.4$, zero interband coupling, $T=0.31T_c$ and $q=0.25$ is simulated.
• Figure 2: Fractional vortex in a $U(1)\times Z_2$ symmetric three-component model. The total magnetic flux in the system is $\Phi_0/3$. Model parameters are the same, as in FIG. \ref{['fig:fv_NJ']}, except interband coupling $V_{12}=V_{13}=V_{23}=-0.6$ and temperature $T=0.27T_c$. On the domain wall, away from the vortex, partial currents compensate, and the total current is zero.
• Figure 3: Domain wall energy in bandwidth units as a function of temperature. A domain wall was generated in the middle of a square sample with $24\times24$ nodes. Results were obtained for slightly asymmetric intraband coupling $V_{11}=2.5$, $V_{22}=V_{33}=2.4$ and various interband couplings $V_{12}=V_{13}=V_{23}=-u$, $u=0.6,\,0.3,\,0.1$. The energy is calculated for both kinds of domain walls - high energy (HE) and low energy (LE). A characteristic temperature exists for each coupling strength, above which the HE domain wall becomes unstable on our numerical grid. For higher temperatures, the domain wall width becomes comparable to the system size.
• Figure 4: Composite single-quantum vortex (consisting of very slightly offset fractional vortices) on top of the domain wall in a $U(1)\times Z_2$ asymmetric three-component model. The total magnetic flux in the system is $\Phi_0$. Model parameters are the same, as in FIG. \ref{['fig:fv_NJ']}, except interband coupling $V_{12}=V_{13}=-0.6$, $V_{23}=-0.5$ and temperature $T=039T_c$.
• Figure 5: Tunneling conductance Eq. (\ref{['eq:tunn_cond']}) for different domain walls and uniform solution (UNI) in the absence of vortices. Simulation parameters are the same as on FIG. \ref{['fig:dw_energy']}, $T=0.25T_c$. For weak interband coupling $s+is$, the domain wall gives nearly no signature in tunneling conductance.