Nodal lines and nodal loops in nonsymmorphic odd-parity superconductors

We discuss the nodal structure of odd-parity superconductors in the presence of nonsymmorphic crystal symmetries, both with and without spin-orbit coupling, and with and without time-reversal symmetry. We comment on the relation of our work to previous work in the literature, and also the implications for unconventional superconductors such as ${\mathrm{UPt}}_3$.

Figure 1

Illustration of nodal loops in ${\mathrm{UPt}}_3$ based on the toy model of Yanase [8]. Solid curves indicate the superconducting state dispersion, and dashed curves the normal state dispersion. (a) Dispersion along $A\text{−}H$ for $\mathrm{\Delta }=0.1$, where $A$ is (0,0,$\pi /c$) and $H$ is ($0,4/3a,\pi /c$). (b) dispersion along $A\text{−}L$ for $\mathrm{\Delta }=0.1$, where $A$ is (0,0,$\pi /c$) and $L$ is ($2\pi$/$\sqrt{3}a,0,\pi /c$). Here, $\mathrm{\Delta }$ is the value of the superconducting $E_{2u}$ order parameter in these energy units, this being the $f$ function of Yanase which pairs electrons between two near-neighbor uranium sites (taken here as a constant for illustrative purposes). The nodes in (a) (due to the absence of intraband pairing for $E_{2u}$ symmetry) and their lack thereof in (b) (due to interband pairing, which is allowed for this symmetry) lead to the two nodal lines closing to form nodal loops in the $k_z=\pi /c$ zone face. (c) Same as (a), but for $\mathrm{\Delta }=0.5$, showing the disappearance of the nodes along $A\text{−}H$, and thus the collapse of the nodal loops [23]. (d) dispersion along $k_z$ normal to the second node along $A\text{−}H$ in (a), illustrating that these are nodal loops, and not toroidal Fermi surfaces. This can also be seen from plots like in (a), where the nodes lift when $k_z$ deviates from $\pi /c$.

Figure 2

Nodal loops in the $k_z=\pi /c$ zone face using the parameters from Fig. 1. They form due to the energy gap from interband pairing that occurs along the $A\text{−}L$ lines.