Topology of Andreev bound states with flat dispersion

Theory of dispersionless Andreev bound states on surfaces of time-reversal-invariant unconventional superconductors is presented. The generalized criterion for the dispersionless Andreev bound state is derived from the bulk-edge correspondence, and the chiral spin structure of the dispersionless Andreev bound states is argued from which the Andreev bound state is stabilized. We then summarize the criterion in a form of index theorems. The index theorems are proved in a general framework to certify the bulk-edge correspondence. As concrete examples, we discuss the (i) $d_{\mathit{xy}}$-wave superconductor, (ii) $p_x$-wave superconductor, and (iii) noncentrosymmetric superconductors. In the last example, we find a peculiar time-reversal-invariant Majorana fermion. The time-reversal-invariant Majorana fermion shows an unusual response to the Zeeman magnetic field, which can be used to identify it experimentally.

Figure 1

Surfaces of superconductors: (a) two-dimensional surface, (b) one-dimensional edge. The conjugate momenta of ${\mathit{x}}_{\perp }$ and ${\mathit{x}}_{\parallel }$ are ${\mathit{k}}_{\perp }$ and ${\mathit{k}}_{\parallel }$, respectively.

Figure 2

The integral path $C$ in the first Brillouin zone: (a) two-dimensional case, (b) three-dimensional case. On the path $C$, the momentum $\mathit{k}$ has a fixed ${\mathit{k}}_{\parallel }$.

Figure 3

Zero energy states of ${\mathcal{H}}^2({\mathit{x}}_{\perp },{\mathit{k}}_{\parallel })$: (a) $N_0^{(+)}=2$ and $N_0^{(-)}=1$, (b) $N_0^{(+)}=1$ and $N_0^{(-)}=0$. By perturbation, some of the zero modes in (a) might become nonzero modes as in (b), however, $N_0^{(+)}$ $-N_0^{(-)}$ does not change.

Figure 4

Fermi surfaces with simple topology in (a) quasi-one-dimensional, (b) quasi-two-dimensional, and (c) three-dimensional systems. The thick blue lines denote the integral path of $w$. For simplicity, we illustrate the integral path only in (a) and (b). For each case, the integral path gets across the Fermi surface only twice at $k_x=\pm k_x^0$. In (a) and (b), $P$ and $P^{\prime }$ denote the intersection points $(-k_x^0,k_y)$ and $(k_x^0,k_y)$, respectively.

Figure 5

The multiple Fermi surfaces in the first Brillouin zone. We take $t=t^{\prime }=1$ and $\mu =-2.5$ in (44).

Figure 6

The energy spectra with open edges at $i_x=0$ and $100$ (upper panels) and the corresponding gap functions in the first Brillouin zone (lower panels) for superconductors with multiple Fermi surfaces. (a) $d_{\mathit{xy}}$-wave superconductor with $\Delta (\mathit{k})={\Delta }_0\sin k_x\sin k_y$, (b) $p_x$-wave superconductor with $\Delta (\mathit{k})={\Delta }_0\sin k_x$, and (c) $p_x$-wave superconductor with $\Delta (\mathit{k})={\Delta }_0\sin 2k_x$. We take $t=t^{\prime }=1$ and $\mu =-2.5$. ${\Delta }_0$ is chosen as ${\Delta }_0=1$ for (a) and ${\Delta }_0=0.5$ for (b) and (c), respectively. In the upper panels, $k_y$ is the momentum in the $y$ direction, and $k_y\in [-\pi ,\pi ]$. The ABSs appear as zero energy states with flat dispersion. In the lower panels, the solid curves denote the Fermi surfaces. The darker areas denote the negative gap functions, the lighter areas the positive ones, and the gap functions vanish at the dashed lines. As a matter of convenience, we take the horizontal axis as the $k_y$ direction in the lower panels.

Figure 7

Fermi surfaces of two-dimensional Rashba noncentrosymmetric superconductors.

Figure 8

(a) A semi-infinite superconductor on $x>0$. The thick curve denotes the confining potential $V(x)$ and the thin curve a gap of the system in the classical limit. If Eq. (87) is satisfied, the gap $\Delta E$ in the classical limit closes near the edge at $x=0$. (b) The corresponding normal dispersion of electrons. Inside the superconductor $(x>0)$, the system supports the Fermi surface, but outside the superconductor $(x<0)$, the system becomes a band insulator. Correspondingly, $\Delta E$ is a superconducting gap for $x>0$, and $\Delta E$ is a band gap [$~V(x)$] for $x<0$.

Figure 9

A zero $(x^0,k_x^0)$ of ${\lambda }_n$ near the edge of a semi-infinite superconductor on $x>0$.

Figure 10

A zero $(x^0,k_x^0)$ of ${\lambda }_n$ near the edge of a semi-infinite superconductor on $x<0$.

Figure 11

Examples of $\Delta (\mathit{k})$ as a function of $k_x$. The symbols $\pm$ represent the sign of $\partial {}_{k_x}\Delta (k_x)$ at $k_x=k_x^0$ with $\Delta (k_x^0)=0$. (a) $\Delta (\mathit{k})$ in a lattice model. From the periodicity in $k_x$, Eq. (B10) holds. (b), (c) $\Delta (\mathit{k})$ in a continuum model. Here we regulate $\Delta (\mathit{k})$ as $\Delta (\mathit{k})\to 0$ at $k_x=\pm \infty$. In the case (c), we need to take into account the contribution at $k_x=\infty$ to obtain Eq. (B10).