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

Abstract

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. We develop a general theory for such a pairing state, which we show to be nonunitary. A time-reversal-odd component of the nonunitary gap product is found to be essential for the appearance of Bogoliubov Fermi surfaces. These Fermi surfaces are topologically protected by a Z2 invariant. We examine their appearance in a generic low-energy effective model and then study two specific microscopic models supporting Bogoliubov Fermi surfaces: 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. The appearance of Bogoliubov Fermi surfaces is accompanied by a magnetization of the low-energy states, which we connect to the time-reversal-odd component of the gap product. 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 part of the gap product as a composite order parameter which is intertwined with superconductivity.

Bogoliubov Fermi surfaces stabilized by spin-orbit coupling

“…Note that SOC lifts the four-fold degeneracy of the j = 3/2 manifold away from the Γ point. Due to the presence of time-reversal and inversion symmetry, the bands remain doubly degenerate so that the states in each band can be labeled by a pseudospin-1/2 index [3].…”

“…In the absence of the pseudomagnetic field, a node occurs where the square root vanishes but the pseudomagnetic field is generally nonzero at these momenta. This lifts the pseudospin degeneracy by shifting the pseudospin-up and pseudospin-down bands in oppsite directions and leads to the formation of BFSs [2,3]. Although this increases the free energy of the TRSB state, for sufficiently small |h k,± | it should not cause a transition to a TRS-preserving phase, since the energy difference between the lowest TRSB and TRS-preserving states is generically finite.…”

“…This permits the construction of novel "internally anisotropic" pairing states, where the Cooper-pair wavefunction has nontrivial dependence upon the orbital or sublattice indices [3,7,8]. Crucially for the appearance of BFSs, internally anisotropic pairing states are typically characterized by both intraband and interband pairing potentials [3]. Such pairing states have been proposed for a wide variety of multiband systems of current interest, such as iron-based superconductors [9][10][11][12][13][14], Cu x Bi 2 Se 3 [15], half-Heusler compounds [2,3,[16][17][18][19][20][21][22][23], the antiperovskite Sr 3−x SnO [24], Sr 2 RuO 4 [25], UPt 3 [26,27], * henri.menke@gmail.com † carsten.timm@tu-dresden.de ‡ philip.brydon@otago.ac.nz transition metal dichalcogenides [28,29], and twisted bilayer graphene [30][31][32].…”

“…It is thus unclear if BFSs can be realized in a limit where they have a detectable effect on the electronic structure [34]. Another interesting question raised by the analysis in [2,3] is what happens at SOC strengths insufficient for a stable TRSB state.…”

Quantitative theory of triplet pairing in the unconventional superconductor LaNiGa2

“…The Cooper pair wave function is symmetric in the crystal momentum and spin channels but it is anti-symmetric with respect to the orbital degree of freedom. Recent studies [11,[30][31][32][33][34][35][36][37][38] in several materials, including the Iron based superconductors, half-Heusler compounds, UPt 3 and Sr 2 RuO 4 , have also pointed out the importance of internal degrees of freedom of electrons (coming from, for example, sublattice or multiple orbitals) in determining the pairing symmetries of superconducting ground states.…”