Symmetries, length scales, magnetic response, and skyrmion chains in nematic superconductors

Nematic systems are two component superconductors that break rotational symmetry, but exhibit a mixed symmetry that couples spatial rotations and phase difference rotations. We show that a consequence of this induced spatial anisotropy is mixed normal modes, that is the linear response to a small perturbation of the system about its ground state, generally couples magnetic and condensate degrees of freedom. We will study the effect of mode mixing on the magnetic response of a nematic system as the strength of applied field is increased. In general we show that the coupled modes generate magnetic field perpendicular to the applied field, causing the magnetic response to spontaneously twist direction. We will study this for the Meissner effect with weak fields and also for stronger applied fields, which produce a mixture of skyrmions and composite vortices, forming orientation dependent bound states. We will also calculate the anisotropies of the resulting first and second critical fields $H_{c_1}$ and $H_{c_2}$. The skyrmion lattices for $H_{c_1}\le H\le H_{c_2}$ in nematic superconductors are shown to be structurally complicated, in contrast to the triangular or square vortex lattices in conventional superconductors. For low fields the magnetic response of the system involves a loosely bound collection of parallel skyrmion chains. As the external field is increased the chains attract one another, causing a transition where the unit cell becomes triangular for high applied fields. This unique skyrmion lattice and the magnetic twisting are clear indicators that could be used experimentally to identify materials that exhibit nematic superconductivity. To obtain these results we develop and present a method to find the unit cell of a vortex lattice that can be applied to other kinds of superconducting systems.

Figure 1

Plot of the linear modes and length scales in the basal plane with spatial dependence ${\widehat{x}}_1=(cos\theta ,sin\theta ,0)$, for successive values of $\eta$ and ${\delta }_{12}=0$. Note that rotations $\theta$ are equivalent to rotating ${\delta }_{12}$ by $2\theta$ instead, due to Eq. (13). The top panels plot the masses (inverse length scales) ${\mu }_i=\frac{1}{{\lambda }_i}$. Each ${\mu }_i$ is a different color, with the real part plotted as a solid line and the imaginary part as a dashed line of the same color. The bottom panel plots the mixing angle ${\theta }_m^i$ of each mode, where the colors of the modes match the colours of the corresponding mass above. The dominant mode is the highest solid line in the top plot and maximum coupling occurs at $\pi /4$ for the bottom plot. So for example, for small $\theta , \eta =0.8$ the dominant mode exhibits strong mixing.

Figure 2

Plot of the linear modes and length scales on a half plane with spatial dependence ${\widehat{x}}_1=(cos\theta ,sin\theta ,1)/\sqrt{2}$, for successive values of $\eta$ and ${\delta }_{12}=0$. Note that rotations $\theta$ are equivalent to rotating ${\delta }_{12}$ by $2\theta$ instead, due to Eq. (13). The top panels plot the masses (inverse length scales) ${\mu }_i=\frac{1}{{\lambda }_i}$. Each ${\mu }_i$ is a different color, with the real part plotted as a solid line and the imaginary part as a dashed line of the same color. The bottom panel plots the mixing angle ${\theta }_m^i$ of each mode, where the colours of the modes match the colours of the corresponding mass above.

Figure 3

Contour plots of the mixing angle ${\theta }_m^1$ given in Eq. (39), corresponding to the leading mode, for ${\delta }_{12}=0$. The position corresponds to the $(x,y)$ components of the normal unit three-vector ${\widehat{x}}_1\in S^2$, where we have projected the northern hemisphere of $S^2$ to a flat disk. Due to reflection symmetry the southern hemisphere will match. Note that changing ${\delta }_{12}$ is equivalent to rotating the plot around the origin by ${\delta }_{12}/2$.

Figure 4

Plot of the Meissner state, with an external field of strength $H_0=1$ and $\eta =1.6$. The applied field is applied in the three directions ${\widehat{x}}_3=(0,1,0)$ (red), ${\widehat{x}}_3=(0,0,1)$ (green), and ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ (blue). We have plotted the total magnetic field $\left|B\right|$, magnetic field parallel to the applied magnetic field $B_3$ and orthogonal $B_2$, as well as the angle between the total magnetic field and the applied field (twisting angle). Finally we have plotted the normalized free energy density, condensate magnitudes ${\rho }_1$ and ${\rho }_2$, and the phase difference ${\theta }_{12}$.

Figure 5

Plot of the Meissner state with boundary normal ${\widehat{x}}_1=(1,0,1)/\sqrt{2}$ and external field strength $H_0=0.3, \eta =1.0$. The applied field is applied in the three directions ${\widehat{x}}_3=(-1,0,1)/\sqrt{2}$ (red), ${\widehat{x}}_3=(0,1,0)$ (green), and ${\widehat{x}}_3=(-\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2})$ (blue). We have plotted the total magnetic field $\left|B\right|$, magnetic field parallel to the applied magnetic field $B_3$ and orthogonal $B_2$, as well as the angle between the total magnetic field and the applied field (twisting angle). Finally we have plotted the normalized free energy density, condensate magnitudes ${\rho }_1$ and ${\rho }_2$, and the phase difference ${\theta }_{12}$.

Figure 6

Contour plots of degenerate single vortex/antivortex solutions in the basal plane ${\widehat{x}}_3=(0,0,1)$. The top row fixes the boundary phase difference to be ${\delta }_{12}=0$, while the bottom row fixes it to be ${\delta }_{12}=\frac{\pi }{2}$. For each solution we have plotted the (normalized) free energy density, magnetic flux $B_3$, condensate magnitudes ${\rho }_1$ and ${\rho }_2$, and the phase difference ${\theta }_{12}$. The solutions form skyrmions such that $Q=1$. Note that there is no in plane magnetic field in the basal plane $B_1=B_2=0$.

Figure 7

Contour plot of $n=2-4$ bound states for $\eta =3$, in the basal plane ${\widehat{x}}_3=(0,0,1)$ with boundary phase difference value ${\delta }_{12}=0$. The normalized energies of the solutions are ${\widehat{F}}_1=7.42, {\widehat{F}}_2=7.30, {\widehat{F}}_3=7.33$, and ${\widehat{F}}_4=7.28$. The solutions exhibit fractional vortex splitting or skyrmions ($\mathcal{Q}=n$), that form bound states.

Figure 8

Contour plots for the $n=1-4$ solutions for $\eta =1$, in the basal plane ${\widehat{x}}_3=(0,0,1)$ with boundary phase difference value ${\delta }_{12}=0$. The solutions do not form skyrmions ($Q=0$), but do exhibit magnetic field inversion.

Figure 9

Contour plot for $\eta =3$, in the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. The coordinates are ${\widehat{x}}_1=(1,0,0)$ and ${\widehat{x}}_2=(0,-1,1)/\sqrt{2}$. Note that spontaneous in-plane magnetic field is generated from the coupled modes, causing the magnetic field to twist significantly.

Figure 10

Contour plot of an $n=2$ bound state for $\eta =3$, in the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. The coordinates are ${\widehat{x}}_1=(1,0,0)$ and ${\widehat{x}}_2=(0,-1,1)/\sqrt{2}$. Note that spontaneous in-plane magnetic field is generated from the coupled modes, causing the magnetic field to twist.

Figure 11

Contour plot of an $n=3$ bound state for $\eta =3$, in the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. The coordinates are ${\widehat{x}}_1=(1,0,0)$ and ${\widehat{x}}_2=(0,-1,1)/\sqrt{2}$. Note that spontaneous in-plane magnetic field is generated from the coupled modes, causing the magnetic field to twist.

Figure 12

Contour plot of an $n=4$ bound state for $\eta =3$, in the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. The coordinates are ${\widehat{x}}_1=(1,0,0)$ and ${\widehat{x}}_2=(0,-1,1)/\sqrt{2}$. Note that spontaneous in-plane magnetic field is generated from the coupled modes, causing the magnetic field to twist.

Figure 13

Contour plots for $n=1-4$ bound state for $\eta =1$, in the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. The coordinates are $x_1=(1,0,0)$ and $x_2=(0,-1,1)/\sqrt{2}$. Note that spontaneous in pane magnetic field are generated from the coupled modes, causing the magnetic field to twist significantly.

Figure 14

Contour plots for $n=1-4$ bound state for $\eta =1$, in the half plane ${\widehat{x}}_3=(1,0,1)/\sqrt{2}$ with boundary phase difference value ${\delta }_{12}=0$. Note that spontaneous in pane magnetic field are generated from the coupled modes, causing the magnetic field to twist significantly.

Figure 15

A plot comparing $H_{c_1}$ (lower critical field) and $H_{c_2}$ (upper critical field) for the basal plane.

Figure 16

The upper critical field $H_{c2}$ as a function of orientation for applied fields directed along ${\widehat{x}}_3=(cos\theta cos\varphi ,cos\theta sin\varphi ,sin\theta )$. A symmetry of the model implies that $H_{c2}$ is independent of $\varphi$.

Figure 17

Diagram of the geometry of a general unit cell of a vortex lattice. The cell is defined by the two vectors $v_1$ and $v_2$.

Figure 18

Contour plots of the vortex lattice for $\eta =3$ in the basal plane ${\widehat{x}}_3=(0,0,1)$ with increasing external field strength $H=H_0{\widehat{x}}_3$. Increasing $H_0$ initially causes the skyrmion chains to squash together. Then the length of the chains is squashed. Finally there is a phase transition to a standard triangular lattice. The unit cells are marked by black lines and tessellated in the plane. Note that the deviation in the order parameter and magnetic field is decreasing for high external field.

Figure 19

Plot of the geometry of the minimal unit cell for $\eta =3$ in the basal plane ${\widehat{x}}_3=(0,0,1)$ for increasing external field strength $H=H0x3$. The top plot shows the relative lengths of the vectors $v_1, v_2$ that form the unit cell. $G/\mathcal{A}$ is the normalized Gibbs free energy per area, which is approximately 0 at $H0=Hc1$ (dashed line). The third plot gives the magnetic response, which is proportional to the inverse of the area $B/\mathcal{A}=2\pi n/\mathcal{A}$ where the winding is initially $n=2$, but for the top row is $n=1$. Finally $\alpha$ is the angle of the unit cell.

Figure 20

Contour plots of the vortex lattice for $\eta =3$ on the half plane ${\widehat{x}}_3=(0,1,1)/\sqrt{2}$ for increasing external field strength $H=H_0{x}_3$. $H_0$ (noted on the left) increases up the page causing the chains to squash together. The unit cells are marked by grey lines and tessellated in the plane. The unit cell has degree $n=2$ and Skyrme charge $Q=2$. At $H_0=3.25$ the skyrmions disappear ($Q=0$), leaving composite vortices. The coordinates are $x_1=(1,0,0), x_2=(0,1,-1)/\sqrt{2}$.