Orbital angular momentum in a topological superconductor with Chern number higher than 1

We investigate the bulk orbital angular momentum (AM) in a two-dimensional hole-doped topological superconductor (SC) which is composed of a hole-doped semiconductor thin film, a magnetic insulator, and an $s$-wave SC and is characterized by the Chern number $C=-3$. In the topological phase, $L_z/N$ is strongly reduced from the intrinsic value by the non-particle-hole-symmetric edge states as in the corresponding chiral $f$-wave SCs when the spin-orbit interactions (SOIs) are small, while this reduction of $L_z/N$ does not work for the large SOIs. Here $L_z$ and $N$ are the bulk orbital AM and the total number of particles at zero temperature, respectively. As a result, $L_z/N$ is discontinuous or continuous at the topological phase transition depending on the strengths of the SOIs. We also discuss the effects of the edge states by calculating the radial distributions of the orbital AM.

Figure 1

Phase diagrams in terms of the Chern number for (a) $|{\mathrm{\Delta }}_{\mathrm{H}}/W|=|{\mathrm{\Delta }}_{\mathrm{L}}/W|=1$ and (b) $|{\mathrm{\Delta }}_{\mathrm{H}}/W|=|{\mathrm{\Delta }}_{\mathrm{L}}/W|=0.35$.

Figure 2

Spectra for (a) $h=0.2<h_{\mathrm{cH}}$ ($C=0$), (b) $h_{\mathrm{cH}}<h=1<h_{\mathrm{cL}}$ ($C=-3$), and (c) $h=2>h_{\mathrm{cL}}$ ($C=-4$). We put $\alpha =1.5$.

Figure 3

(a) $h$ and (b) $\alpha$ dependences of the total spin $J_z/N$ and the bulk orbital AM $L_z/N$. Symbols and lines are obtained by the real and reciprocal spaces, respectively. In (a), the light and dark gray regions indicate the topological phases with $C=-3,-4$, respectively.

Figure 4

(a) $h$ and (b) $\alpha$ dependences of the Fermi AM. For $C=-4$, only the outer Fermi AM is shown because the inner Fermi AM is found to be small. In (a), the light and dark gray regions indicate the topological phases with $C=-3,-4$, respectively.

Figure 5

Black solid line indicates the band structure for the BdG Hamiltonian Eq. (1), while red broken and blue dotted lines indicate the electron and hole bands in the normal state, respectively. We set $h=0.5$ and (a) $\alpha =0.5$, (b) $\alpha =0.8$, (c) $\alpha =0.9$, (d) $\alpha =1.2$. Between (b) and (c), the Fermi AM shows the sharp decrease in Fig. 4.

Figure 6

Radial distributions of (a) ${\ell }_z^{\left(\mathrm{r}\right)}\left(r\right)$, (b) ${\ell }_z^{i=1}\left(r\right)$, and (c) ${\ell }_z^{i\ne 1}\left(r\right)$ for $h<h_{\mathrm{c}}=0.61$ ($C=0$) and $h>h_{\mathrm{c}}$ ($C=-1$) in an electron-doped topological SC. We set $\alpha =0.1$. See Ref. [44] for the other parameters.

Figure 7

Radial distributions of (a) ${\ell }_z^{\left(\mathrm{r}\right)}\left(r\right)$, (b) ${\ell }_z^{i=2}\left(r\right)$, (c) ${\ell }_z^{i=1}\left(r\right)$, and (d) ${\ell }_z^{i\ne 1,2}\left(r\right)$ for $h<h_{\mathrm{cH}}=0.47$ ($C=0$) and $h>h_{\mathrm{cH}}$ ($C=-3$) in a hole-doped topological SC. We set $\alpha =0.7$.