Disorder-induced exceptional points and nodal lines in Dirac superconductors

We consider the effect of disorder on the spectrum of quasiparticles in the point-node and nodal-line superconductors. Due to the anisotropic dispersion of quasiparticles disorder scattering may render the Hamiltonian describing these excitations non-Hermitian. Depending on the dimensionality of the system, we show that the nodes in the spectrum are replaced by Fermi arcs or Fermi areas bounded by exceptional points or exceptional lines, respectively. These features are illustrated by first considering a model of a proximity-induced superconductor in an anisotropic two-dimensional (2D) Dirac semimetal, where a Fermi arc in the gap bounded by exceptional points can be realized. We next show that the interplay between disorder and supercurrents can give rise to a 2D Fermi surface bounded by exceptional lines in three-dimensional (3D) nodal superconductors.

Figure 1

The lines of zeros in the spectrum of quasiparticles of 2D Dirac semimetal with tilted Dirac cone and proximity-induced superconductivity. The plots show the zero solutions for the spectrum of quasiparticles, given explicitly by Eq. (4) in the text. Panels (a) and (b) show the solutions at two tilt parameters $\kappa =0.4$ and $\kappa =1.5$, respectively, for a fixed $\mathrm{\Delta }$ and the chemical potential $\mu =4\mathrm{\Delta }$. The nodal lines exist at small tilt parameter $\mathrm{\Delta }/\mu <\left|\kappa \right|<1$. Panels (c) and (d) show the nodal lines in the spectrum at $\mu =0.5\mathrm{\Delta }$ and $\mu =0$, respectively, at fixed $\mathrm{\Delta }$ and $\kappa =1.5$.

Figure 2

Panel (a) shows the gapped real part of the spectrum $E\left(\mathbf{k}\right)$ given by Eq. (10) at $k_x=0$ as a function of momentum $k_y$ at ${\mathrm{\Gamma }}_2/\left|\kappa \right|<\mathrm{\Delta }$. Panel (b) shows the spectrum at the phase transition point ${\mathrm{\Gamma }}_2/\left|\kappa \right|=\mathrm{\Delta }$, at which the superconducting gap closes and then transforms into two exceptional points. Panels (c) and (d) show respectively the real and imaginary parts of the spectrum at ${\mathrm{\Gamma }}_2/\left|\kappa \right|>\mathrm{\Delta }$.

Figure 3

Top: Scheme of the Josephson junction through 2D states of the Dirac semimetal. We assume the condition $L_y\ll \frac{v\left|\right(\left|\kappa \right|-1\left)\right|}{\mathrm{\Delta }}\ll L_x$ is met. Bottom: Spectrum for the two-dimensional Dirac semimetal with proximity induced superconducting gap from a numerical diagonalization of the discretized “non-Hermitian” Hamiltonian, $H+\mathrm{\Sigma }=\kappa sin{k}_x+[{\sigma }_{x}sin{k}_y-{\sigma }_y(sin{k}_x+i\gamma )]{\tau }_z+(2-cos{k}_x-cos{k}_y){\sigma }_z-i\mathrm{\Gamma }+\mathrm{\Delta }{\tau }_x$, as a function of $k_x$ for a sample of finite width in the $y$ direction (here the energy is measured in the units of nearest-neighbor hopping energy, and the distance in units of the lattice constant). (a)–(c) Two one-way propagating chiral modes and the bulk modes, where (a) describes weak tilt, while (b) and (c) describe strong tilt cases. The increase of the scattering strength merges surface modes with the bulk. (d) The imaginary part of the dispersion. The top and bottom flat regions correspond to imaginary part of the two surface state's dispersion, which directly connect with the bulk modes. Parameters are chosen as follows: $\mathrm{\Delta }=0.3$, (a) $\kappa =0.9$, $\mathrm{\Gamma }=0.2$, and $\gamma =0.18$; (b) $\kappa =1.2$, $\mathrm{\Gamma }=0.12$, and $\gamma =0.1$; (c), (d) $\kappa =1.2$, $\mathrm{\Gamma }=4.8$, and $\gamma =0.4$.

Figure 4

Top: momentum resolved spectral function $A(\mathbf{k},0)$ at zero energy for the cases of (a) strong proximity effect, in which $\mathrm{\Delta }/{\mathrm{\Gamma }}_2\sqrt{1+{\kappa }^{-2}}\approx 4>1$, and (b) weak proximity effect ($\mathrm{\Delta }/{\mathrm{\Gamma }}_2\sqrt{1+{\kappa }^{-2}}\approx 0.4<1$) for fixed $\left|\kappa \right|=1.3$. The thick light green lines indicate a large spectral weight. One sees the emergence of the bulk Fermi arc as the value of the proximity induced gap $\mathrm{\Delta }$ decreases. Bottom: spectral function $A(0,E)$ at zero momentum for the cases of strong (c) and weak (d) proximity effect, which can be distinguished by the number of peaks in the spectral function.

Figure 5

Panels (a) and (b) show respectively the real and imaginary parts of the spectrum $E\left(\mathbf{k}\right)$, given by Eq. (14), as a function of momentum $k_y$ at $k_x=k_z=0$ for the disordered nodal-ring superconductor (or polar phase of the superfluid ${}^3\mathrm{He}$) with the supercurrent applied along the $z$ axis. The interplay of the supercurrent and disorder splits the nodal-ring into a region where the real part of the spectrum is zero, bounded with the outer and inner exceptional circumferences. The spectrum has a square root singularity at the exceptional rings.