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 CuₓBi₂Se₃. This suggests that the pairing symmetry corresponds to a two-dimensional representation of the D3d crystal point group, and that CuₓBi₂Se₃ is a nematic superconductor. In this paper, we present a comprehensive study of the upper critical field Hc2 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 D3d crystals reflects the true crystal rotation symmetry. This has direct implications for the upper critical field. First, Hc2 of trigonal superconductors with D3d symmetry exhibits a sixfold anisotropy in the basal plane. Second, when the degeneracy of the two components is lifted by, e.g., uniaxial strain, Hc2 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 CuₓBi₂Se₃, SrₓBi₂Se₃, NbₓBi₂Se₃, and TlₓBi₂Te₃.