Gapless topological superconductors: Model Hamiltonian and realization

The existence of an excitation gap in the bulk spectrum is one of the most prominent fingerprints of topological phases of matter. In this paper, we propose a family of two-dimensional Hamiltonians that yield an unusual class $D$ topological superconductor with a gapless bulk spectrum but well-localized Majorana edge states. We perform a numerical analysis for a representative model of this phase and suggest a concrete physical realization by analyzing the effect of magnetic impurities on the surface of a strong topological insulator.

Figure 1

(a) Phase diagram as a function of Zeeman field and disorder strength. Phase transitions are shown in red. $M$ denotes a metallic phase which typically forms in a disordered two-dimensional superconductor [16]. (b) Disorder averaged bulk thermal conductance of the model in Eq. (4). See Ref. [17] for simulation parameters.

Figure 2

The Fermi surface that arises from $H$: (a) the blue curves denote regions of the outer Fermi surface that will be gapped by the magnetization and the red curves denote regions that will be gapped by superconductivity. (b) $H_{\mathrm{m}}$ defines a new Brillouin zone and leads to an open Fermi surface. The addition of superconductivity gaps out the remaining Fermi surface except for four discrete points, as shown in Fig. 3.

Figure 3

Band structure of the Hamiltonian, Eq. (2), with (a) periodic boundary conditions. The red and blue regions are gapped by the superconductor and the magnetization, respectively (cf. Fig. 2). (b) An edge along the $y$ direction. Gapless edge states (red, green) coexist with the bulk nodes (blue). (c) Typical edge state wave functions.

Figure 4

(a) The spectrum of a system with an edge along the $y$ direction for all possible terminations. The colors in the legend denote the right edge state. (b) Strong edge contribution, $G_{\mathrm{ch}}=G_{RL}-G_{LR}$, to the thermal conductance as a function of disorder strength $\delta$. We use $m=0.1$ and $\mathrm{\Delta }=0.05$. The chirality of the edge states changes depending on the termination and on disorder.

Figure 5

Edge contribution, $G_{\mathrm{ch}}$, to the thermal conductance in the absence of disorder ($\delta =0$) as a function of system size for the different terminations. Here, $m=0.1$ and $\mathrm{\Delta }=0.05$. (a) Strong edge states. While their chirality depends on the termination, their contribution to the conductance approaches unity. (b) Weak edge states. Independently of the termination, their contribution approaches zero.

Figure 6

(a) Setup illustration and the spiral spin order in the ground state for an aligned hexagonal lattice of the magnetic impurities. (b) Stability of the spiral order $\langle \rho \rangle$ against the normalized inverse temperature $\tilde{\beta }$; both defined in the text.

Figure 7

Sketch of the three-terminal geometry used in transport simulations. The edge states (arrows) give the only chiral contribution to transport.

Figure 8

Illustration of the connection of our findings to Sau's model [22] in the limit of small $Q$. Upper part: the system, in real space, is described as stripes of superconductors, with alternating Chern numbers, connected in parallel. The chiral Majorana edge states of each stripe are also depicted. Lower part: the gapped bulk states of each stripe are ignored, and the system may be thought of as a collection of one-dimensional chiral channels connected in parallel and coupled by nearest-neighbor coupling.

Figure 9

Fermi surfaces of ${\mathcal{H}}_0$ in Eq. (C1) for various chemical potentials. The initially circular Fermi surface becomes hexagonal with the nesting vectors $\pm {\left(Q_i\right)}_{i=1}^3$. The hexagonal regime is marked off by red lines.

Figure 10

The largest eigenvalue of the spin susceptibility $\chi$ in momentum space at zero temperature and for $\mu =0.7$.

Figure 11

The largest eigenvalue of the spin susceptibility $\chi$ for different cutoffs $\mathrm{\Lambda }$ at zero temperature and $\mu =0.7$. For large cutoffs, the concentration of $\chi$ around the nesting momenta gets reduced and the contributions at low momenta have to be taken into account.

Figure 12

The largest eigenvalue of the RKKY interaction in momentum space for different lattice structures and lattice constants, $a$. The borders of the Brillouin zones are marked by dotted lines. A square, hexagonal, and by $\pi /3$ rotated hexagonal lattice, is shown in the top, middle, and bottom panels, respectively.