Identification of nematic superconductivity from the upper critical field

Recent nuclear magnetic resonance and specific heat measurements have provided concurring evidence of spontaneously broken rotational symmetry in the superconducting state of the doped topological insulator ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. This suggests that the pairing symmetry corresponds to a two-dimensional representation of the $D_{3d}$ crystal point group, and that ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ is a nematic superconductor. In this paper, we present a comprehensive study of the upper critical field $H_{c2}$ of nematic superconductors within Ginzburg-Landau (GL) theory. Contrary to typical GL theories which have an emergent U(1) rotational symmetry obscuring the discrete symmetry of the crystal, the theory of two-component superconductors in trigonal $D_{3d}$ crystals reflects the true crystal rotation symmetry. This has direct implications for the upper critical field. First, $H_{c2}$ of trigonal superconductors with $D_{3d}$ symmetry exhibits a sixfold anisotropy in the basal plane. Second, when the degeneracy of the two components is lifted by, e.g., uniaxial strain, $H_{c2}$ exhibits a twofold anisotropy with characteristic angle and temperature dependence. Our thorough study shows that measurement of the upper critical field is a direct method of detecting nematic superconductivity, which is directly applicable to recently-discovered trigonal superconductors ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, ${\mathrm{Sr}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, ${\mathrm{Nb}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{Tl}}_x{\mathrm{Bi}}_2{\mathrm{Te}}_3$.

Figure 1

Upper critical field ($H_{c2}$) anisotropy of two-component pairing in trigonal crystals with $D_{3d}$ point group symmetry, originating from the trigonal GL anisotropy term (4). (a) Polar plot of the angular dependence of $H_{c2}$ with sixfold symmetry given by Eq. (10) (normalized by ${\tilde{H}}_{c2}$) for $J_4/J_1=0.2$. Different curves correspond to $J_5/\omega =J_5/\sqrt{J_1{J}_3}=(0.15,0.30,0.45,0.60)$ (inward to outward). (b) Same as (a) but for $J_4/J_1=0.4$.

Figure 2

(a) Polar plot of the angular dependence of $H_{c2}$ in the presence of a symmetry-breaking field $\delta$ for $J_4/J_1=0.1$, calculated using Eq. (15) (in arbitrary units of $H$). Different curves represent different temperatures: $T=T_c^{*}-t\mathrm{\Delta }{T}_c$ (recall that $\mathrm{\Delta }{T}_c\sim \left|\delta \right|$), where $t=1,...,8$ and the outermost curve corresponds to $t=8$. (b) Same as in (a) but for relatively large $J_4/J_1=0.6$. Figure (b) clearly shows the twofold “peanut”-shape anisotropy expected for two-component superconductors in the presence of a symmetry breaking field. (c) Plot of the $H_{c2}$-anisotropy coefficient $H_{c2}\left(\frac{\pi }{2}\right)/H_{c2}\left(0\right)$ as function of effective temperature $t$ for various values of $J_4/J_1$. The horizontal grid lines correspond to the values $(1+J_4/2J_1)/(1-J_4/2J_1)$.