Magnetic field dependence of the superconducting gap node topology in noncentrosymmetric $\mathrm{Ce}{\mathrm{Pt}}_3\mathrm{Si}$

Noncentrosymmetric superconductors, such as $\mathrm{Ce}{\mathrm{Pt}}_3\mathrm{Si}$ and ${\mathrm{Li}}_2\mathrm{Pt}{\mathrm{B}}_2$, are believed to have a line node in the energy gap arising from the coexistence of $s$-wave and $p$-wave pairing. Using as an example $\mathrm{Ce}{\mathrm{Pt}}_3\mathrm{Si}$, we show that a weak $c$-axis magnetic field will remove this line node, since it has no topological stability against time-reversal symmetry-breaking perturbations. Conversely, a field in the $a\text{−}b$ plane is shown to remove the line node on some regions of the Fermi surface, while bifurcating the line node in other directions, resulting in two “boomerang”-like shapes. These line-node topological changes are predicted to be observable experimentally in the low-temperature heat capacity.

Figure 1

(Color online) Illustration of the influence of the $ab$-plane external magnetic field on the gap line nodes in $\mathrm{Ce}{\mathrm{Pt}}_3\mathrm{Si}$. (a) Zero-field line nodes for $\pm k_z=\mathrm{const}$ on one Rashba Fermi-surface sheet; (b) $H_y\ne 0$ removes the node along the $\pm k_y$ direction and bifurcates the node elsewhere, yielding “boomerang”-like nodal topology.

Figure 2

(Color online) Normal state (left) and superconducting state (right) quasiparticle dispersions ${\epsilon }_k$ and $E_{\mathbf{k}}$ for particles and holes as a function of $k={(k_x^2+k_y^2)}^{1∕2}$ for a fixed $k_z$. From top to bottom, zero field $\mathbf{H}=0$, $\mathbf{H}\perp {\mathbf{g}}_{\mathbf{k}}$, and $\mathbf{H}\Vert {\mathbf{g}}_{\mathbf{k}}$.

Figure 3

(Color online) Calculated temperature dependence of the specific heat with and without Zeeman field along $z$ direction (a) and along $y$ direction (b). The inset shows the low-temperature region for $H_z=0$ and $H_z=3\mathrm{T}$. Here, we assume the BCS-like temperature dependence of both the $p$-wave and $s$-wave components of the superconducting gap with ${\mathbf{d}}_{\mathbf{k}}\left(0\right)=0.2\mathrm{meV}$ and ${\Delta }_{\mathbf{k}}\left(0\right)=0.25{\Delta }_p\left(0\right)$, which gives about the BCS ratio between the magnitude of the superconducting gap and $T_c=0.75\mathrm{K}$. Following experimental findings we adopt $E_f\approx 0.012\mathrm{eV}$ and estimate $k_F\approx 0.5{\mathrm{\AA }}^{-1}$, which are typical values for the heavy-fermion metals. We further approximate the Rashba splitting of the bands $\alpha \sim 1\times {10}^{-3}\mathrm{eV}\mathrm{\AA }$.