Experimental consequences of Bogoliubov Fermi surfaces

Superconductors involving electrons with internal degrees of freedom beyond spin can have internally anisotropic pairing states that are impossible in single-band superconductors. As a case in point, in even-parity multiband superconductors that break time-reversal symmetry, nodes of the superconducting gap are generically inflated into two-dimensional Bogoliubov Fermi surfaces. The detection and characterization of these quasiparticle Fermi surfaces requires the understanding of their experimental consequences. In this paper, we derive the low-energy density of states for a broad range of possible nodal structures. Based on this, we calculate the low-temperature form of observables that are commonly employed for the characterization of nodal superconductors, i.e., the single-particle tunneling rate, the electronic specific heat and Sommerfeld coefficient, the thermal conductivity, the magnetic penetration depth, and the NMR spin-lattice relaxation rate, in the clean limit. We also address the question whether the topological invariant of the Bogoliubov Fermi surfaces is associated with topologically protected surface states, with negative results. This work is meant to serve as a guide for experimental searches for Bogoliubov Fermi surfaces in time-reversal-symmetry-breaking superconductors.

Figure 1

Density of states for various values of the exponent $g=1/m+1/n$.

Figure 2

Temperature-dependent part $\mathrm{\Delta }{\lambda }_i$ of the magnetic penetration depth (blue curve) as a function of temperature $T$ for $k_{B}T$ small compared to the gap amplitude ${\mathrm{\Delta }}_0$. The asymptotic form for $k_{B}T\ll h,{\mathrm{\Delta }}_0$ is shown for comparison (dashed black curve).

Figure 3

Bulk Fermi surfaces in the normal (semitransparent gray) and superconducting (orange) states of the model used in the search for surface states. Due to the weak dispersion in the $z$ direction, the normal-state Fermi surface and two of the BFSs are deformed open cylinders.

Figure 4

Smallest non-negative eigenenergy $E_{\min }$ of the slab Hamiltonian as a function of momentum ${\mathbf{k}}_{\parallel }=(k_x,k_y)$ in the two-dimensional Brillouin zone. Zero-energy states are shown in black. The black regions are the projections of the BFSs. The thickness of the slab is $N=180$ unit cells.

Figure 5

Dispersion of low-lying nonnegative eigenenergies of the slab Hamiltonian (black) along the $k_x=0$ axis in the two-dimensional Brillouin zone. The thickness of the slab is $N=100$. The dispersion is overlaid over the non-negative eigenenergies for periodic boundary conditions and the same thickness (cyan), which mimic the bulk dispersion.