Superconducting gap anisotropy and topological singularities due to lattice translational symmetry and their thermodynamic signatures

Symmetry arguments based on the point group of a system and thermodynamic measurements are often combined to identify the order parameter in unconventional superconductors. However, lattice translations, which can induce additional momenta with vanishing order parameter in the Brillouin zone, are neglected, especially in gap functions otherwise expected to be constant, such as in chiral superconductors. After a general analysis of the symmetry conditions for vanishing gap functions, we study the case of chiral $p$- and chiral $f$-wave pairing on a square lattice, a situation relevant for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Specifically, we calculate the impurity-induced density of states, specific heat, superfluid density, and thermal conductivity employing a self-consistent $T$-matrix calculation and compare our results to the case of a nodal ($d$-wave) order parameter. While there is a clear distinction between a fully gapped chiral state and a nodal state, the strongly anisotropic case is almost indistinguishable from the nodal case. Our findings illustrate the difficulty of interpreting thermodynamic measurements. In particular, we find that the available measurements are consistent with a chiral ($f$-wave) order parameter. Our results help to reconcile the thermodynamic measurements with the overall picture of chiral spin-triplet superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

Example Fermi surfaces for $\mu =-1.0$ (green), $\mu =0.1$ (blue), and $\mu =0.47$ (pink). The black dots mark the symmetry-imposed gap zeros with associated winding numbers for dominant $f$-wave order parameter and negative chirality. For a superposition of $p$-wave and $f$-wave pairs with $|{\mathrm{\Delta }}_p|<|{\mathrm{\Delta }}_f|$, the zeros move along the diagonals towards the BZ corner (center), for equal (different) relative sign.

Figure 2

Gap anisotropy and phase winding for a $p$- and $f$-wave order parameter for different chemical potentials $\mu$; see Fig. 1. For better comparison, the gap magnitude is normalized with respect to ${\mathrm{\Delta }}_p$ and ${\mathrm{\Delta }}_f$, respectively.

Figure 3

The order parameter components and free energy $f$ as a function of the chemical potential $\mu$ derived from the GL theory. The vertical dashed line illustrates the point, where the Chern number changes from 1 to $-3$. (a) The magnitude of the $p$ wave, ${\eta }_p$, in the uniform state. (b) The magnitude of the $f$ wave, ${\eta }_f$, in the uniform state. (c) The phase difference between the $p$-wave and $f$-wave state, $\mathrm{\Delta }\theta$. (d) The free energy normalized by the parameters $a_p^{\prime }{T}_{cp}$.

Figure 4

The relative change of the gap ratio ${\mathrm{\Delta }}_p/{\mathrm{\Delta }}_f$ at $\mu =0.1$ as a function of the impurity concentration $c$. This change is shown for two different combinations for the temperatures $T_c$ and $T_0=T_c/50$, where $T_c$ depends on the impurity concentration. The impurity concentration $c$ is normalized to the critical value $c_c$.

Figure 5

Summary of thermodynamic properties: (a) specific heat, (b) superfluid density, and (c) thermal conductivity as a function of temperature $T$ for a $p$-, $f$-, and $d$-wave state at $\mu =0.1$. The solid (dashed) line illustrates the results obtained for zero (finite) disorder. In the case of finite disorder, the impurity concentration is $c/c_c\approx 0.12$. The superfluid density is normalized to the zero-temperature, zero-impurity constant, $n_{s0}$. In the case of $\kappa$, a finite-impurity concentration is necessary. Therefore, the solid (dashed) lines show results for a concentration of $c/c_c\approx 0.05$ ($c/c_c\approx 0.28$).

Figure 6

The residual thermal conductivity ${\kappa }_0=\kappa (T=0)$ of the $f$-wave state as a function of impurity concentration $c$ ($c_c$ is the critical concentration) at different values of the chemical potential $\mu$. Note that we have used a smaller gap for the computation of this figure compared to the previous ones. This explains the different magnitude of ${\kappa }_0/T$.

Figure 7

The density of states as a function of the angle $\varphi$ and quasiparticle energy $E$ calculated for the $p$-, $f$-, and $d$-wave states at the impurity concentration $c/c_c\approx 0.18$.

Figure 8

The residual thermal conductivity as a function of impurity concentration at $\mu =0.1$ for different superpositions of the $p$ wave and $f$ wave.

Figure 9

Zeros in the gap function and corresponding winding numbers for chiral $d$- and chiral $p$-wave gap functions, as well as a superposition of the two. Note that the situation sketched here corresponds to dominant $d$-wave pairing. The black arrow denotes the direction the zeros move when changing the ratio of the pairing channels.