This is a supplementary material for
arxiv:2003.10917.
See the published version in Phys. Rev. B
Introduction
Non-centrosymmetric superconductors,
with the crystal structure possessing the $O$ point group symmetry,
are described by a single component order parameter
$\psi=|\psi|\mathrm{e}^{i\varphi}$. It is a complex scalar field that is
coupled to the vector potential $\A$ of the magnetic field $\B=\Curl\A$
via the gauge derivative $\D\equiv\Grad-ie\A$. Using $\hbar=c=1$, such
materials are described, in the vicinity of the superconducting critical
temperature, by the Ginzburg-Landau free energy $F=\int {\cal F} d{\bf r}$,
whose density reads as:
\begin{equation*}
{\cal F}= \frac{\B^2}{8\pi}+k|\D\psi|^2+\gamma\j\cdot\B
+\frac{\beta}{2}(|\psi|^2-\psi_0^2)^2 \,,
~~\text{where}~~\j=2e\,\Im\left(\psi^*\D\psi\right)
\,.
\end{equation*}
The explicit breaking of the inversion symmetry is accounted by the Lifshitz
invariant term with the prefactor $\gamma$, that directly couples the magnetic
field $\B$ and the supercurrent $\j$.
The physical length scales of the theory are the coherence length $\xi$ and
the London penetration depth $\lambda_L$, defined respectively as
$\xi^{-2}=2\beta\psi_0/k$, and $\lambda_L^{-2}=8\pi ke^2\psi_0^2$.
Helical streamlines with/without field inversion
The magnetic field forms helicoidal patterns around a straight static vortex,
in a non-centrosymmetric superconductor. For important values of the parity-breaking
coupling $\gamma$, the magnetic field $\B$ can further show inversion patterns
around the vortex. That is, as the distance from the vortex core increases,
the longitudinal component of the magnetic field may change it sign.
The displayed quantities are : On the two static planes, normal with respect to the vortex line, the colors encode the amplitude of the magnetic field $\B$, while the arrows demonstrate the orientation of the field.
The tubes represent streamlines of the magnetic magnetic field between both planes.
The displayed quantities are : On the two static planes, normal with respect to the vortex line, the colors encode the amplitude of the magnetic field $\B$, while the arrows demonstrate the orientation of the field.
The tubes represent streamlines of the magnetic magnetic field between both planes.
Movies 1 & 2 -
Helical structure of the magnetic field streamlines around vortices in
non-centrosymmetric superconductors. The left video shows the helical structure of
the streamlines for moderate value of the parity-breaking coupling $\gamma/k\lambda=0.2$.
The streamlines here feature all the same chirality.
The right panel corresponds to rather important parity-breaking coupling
$\gamma/k\lambda=0.8$, for which the longitudinal component of the magnetic field
is inverted at some distance from the core. The chirality of the streamline
depends on whether the longitudinal component of the magnetic field is inverted.
That is depending on the chirality, some of the streamlines propagate forward
(along positive $z$-direction), while other propagate backward (along negative
$z$-direction)
More details : for vortex with field inversion
Movie 3 -
Magnetic streamlines, emphasizing forward propagating lines (solid tubes),
in the case of field inversion due to important parity-breaking coupling
$\gamma/k\lambda=0.8$. The transparent tubes propagate backward (along the
negative $z$-direction).
Movie 4 -
Magnetic streamlines, emphasizing backward propagating lines (solid tubes),
in the case of field inversion due to important parity-breaking coupling
$\gamma/k\lambda=0.8$. The transparent tubes propagate forward (along the
positive $z$-direction).
References
-
J. Garaud, M. N. Chernodub and D. E. Kharzeev,
Vortices with magnetic field inversion in noncentrosymmetric superconductors.
Phys. Rev. B. 102, 184516 (2020). Link , arXiv -
A. Samoilenka and E. Babaev,
Spiral magnetic field and bound states of vortices in noncentrosymmetric superconductors.
Phys. Rev. B. 102, 184517 (2020). Link , arXiv