Nematic and time-reversal breaking superconductivities coexisting with quadrupole order in a ${\mathrm{\Gamma }}_3$ system

We discuss superconductivity in a model on a cubic lattice for a ${\mathrm{\Gamma }}_3$ non-Kramers system. In previous studies, it is revealed that $d$-wave superconductivity with $E_g$ symmetry occurs in a wide parameter range in a ${\mathrm{\Gamma }}_3$ system. Such anisotropic superconductivity can break the cubic symmetry of the lattice. In a ${\mathrm{\Gamma }}_3$ system, the quadrupole degrees of freedom are active and the effect of the cubic symmetry breaking should be important. Here, we investigate the coexisting states of the $d$-wave superconductivity and quadrupole order by a mean-field theory. In particular, we discuss possible competition and cooperation between the superconductivity and quadrupole order depending on types of them. We find nematic superconductivity breaking the cubic symmetry and coexisting with quadrupole order. In the present model, we also find $d+id$ superconductivity, which breaks time-reversal symmetry but retains the cubic symmetry. We also discuss the effects of uniaxial stress on these superconducting states.

Figure 1

Temperature dependence of the quadrupole order parameters (a) for $J_{20}\left(\mathit{0}\right)=-8t$, (b) for $J_{22}\left(\mathit{0}\right)=-8t$, (c) for $J_{20}\left(\mathit{Q}\right)=-8t$, and (d) for $J_{22}\left(\mathit{Q}\right)=-8t$.

Figure 2

Temperature dependence of the superconducting order parameters (a) for $V_{x^2-y^2}=10t$, (b) for $V_{3z^2-r^2}=10t$, and (c) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$. The induced $O_2^0\left(\mathit{0}\right)$ is also shown.

Figure 3

Phase diagrams of superconductivity coexisting with FQ order: in the $J_{20}\left(\mathit{0}\right)-T$ plane (a) for $V_{x^2-y^2}=10t$, (b) for $V_{3z^2-r^2}=10t$, and (c) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$; in the $J_{22}\left(\mathit{0}\right)-T$ plane (d) for $V_{x^2-y^2}=10t$, (e) for $V_{3z^2-r^2}=10t$, and (f) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$.

Figure 4

Phase diagrams of superconductivity coexisting with AFQ order: in the $J_{20}\left(\mathit{Q}\right)-T$ plane (a) for $V_{x^2-y^2}=10t$, (b) for $V_{3z^2-r^2}=10t$, and (c) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$: in the $J_{22}\left(\mathit{Q}\right)-T$ plane (d) for $V_{x^2-y^2}=10t$, (e) for $V_{3z^2-r^2}=10t$, and (f) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$.

Figure 5

Phase diagram of $d_{x^2-y^2}$ superconductivity coexisting with FQ order of $O_2^0$ for $V_{x^2-y^2}=5t$.

Figure 6

Phase diagram of superconductivity in the $H_{20}-T$ plane without quadrupole interactions for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$.

Figure 7

Phase diagrams of superconductivity coexisting with FQ order in the $H_{20}-T$ plane: with $J_{20}\left(\mathit{0}\right)=-10t$ (a) for $V_{x^2-y^2}=10t$, (b) for $V_{3z^2-r^2}=10t$, and (c) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$; with $J_{22}\left(\mathit{0}\right)=-10t$ (d) for $V_{x^2-y^2}=10t$, (e) for $V_{3z^2-r^2}=10t$, and (f) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$.

Figure 8

Phase diagrams of superconductivity coexisting with AFQ order in the $H_{20}-T$ plane: with $J_{20}\left(\mathit{Q}\right)=-10t$ (a) for $V_{x^2-y^2}=10t$, (b) for $V_{3z^2-r^2}=10t$, and (c) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$; with $J_{22}\left(\mathit{Q}\right)=-10t$ (d) for $V_{x^2-y^2}=10t$, (e) for $V_{3z^2-r^2}=10t$, and (f) for $V_{x^2-y^2}=V_{3z^2-r^2}=10t$.