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 Cu$_x$Bi$_2$Se$_3$. This suggests that the pairing symmetry corresponds to a two-dimensional representation of the $D_{3d}$ crystal point group, and that Cu$_x$Bi$_2$Se$_3$ is a nematic superconductor. In this work, 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 crystals reflects the true $D_{3d}$ crystal symmetry. This has direct implications for the upper critical field. First, $H_{c2}$ of trigonal superconductors 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 Cu$_x$Bi$_2$Se$_3$, Sr$_x$Bi$_2$Se$_3$, Nb$_x$Bi$_2$Se$_3$, and Tl$_x$Bi$_2$Se$_3$.