Bogoliubov Fermi surfaces: General theory, magnetic order, and topology

We present a comprehensive theory for Bogoliubov Fermi surfaces in inversion-symmetric superconductors which break time-reversal symmetry. A requirement for such a gap structure is that the electrons posses internal degrees of freedom apart from the spin (e.g., orbital or sublattice indices), which permits a nontrivial internal structure of the Cooper pairs. In a pairing state that breaks time-reversal symmetry, the Cooper pairs are generically polarized in the internal degrees of freedom, in analogy to spin-polarized pairing in a nonunitary triplet superconductor. We show that this polarization can be quantified in terms of the time-reversal-odd part of the gap product, i.e., the matrix product of the pairing potential with its Hermitian conjugate. This product is essential for the appearance of Bogoliubov Fermi surfaces and their topological protection by a Z(2) invariant. After studying the appearance of Bogoliubov Fermi surfaces in a generic two-band model, we proceed to examine two specific cases: a cubic material with a j = 3/2 total-angular-momentum degree of freedom and a hexagonal material with distinct orbital and spin degrees of freedom. For these model systems, we show that the polarized pairing generates a magnetization of the low-energy states. We additionally calculate the surface spectra associated with these pairing states and demonstrate that the Bogoliubov Fermi surfaces are characterized by additional topological indices. Finally, we discuss the extension of phenomenological theories of superconductors to include Bogoliubov Fermi surfaces, and identify the time-reversal-odd polarization of the Cooper pairs as a composite order parameter, which is intertwined with superconductivity.