Self-consistent study of Abelian and non-Abelian order in a two-dimensional topological superconductor

We perform self-consistent studies of two-dimensional (2D) $s$-wave topological superconductivity (TSC) with Rashba spin-orbit coupling and Zeeman field by solving the Bogoliubov-de Gennes equations. In particular, we examine the effects of a nonmagnetic impurity in detail and show that the nature of the spin-polarized midgap bound state varies significantly depending on the material parameters. Most notably, a nonmagnetic impurity in a 2D $s$-wave topological superconductor can act like a magnetic impurity in a conventional $s$-wave superconductor, leading to phase transitions of the ground state as the impurity potential is varied. Furthermore, by solving for the spin-dependent Hartree potential self-consistently along with the superconducting order parameter, we demonstrate that topological charge density waves can coexist with TSC at half filling just as in a conventional $s$-wave superconductor.

Figure 1

(a) Order parameter ${\mathrm{\Delta }}_0/t$ as a function of the pairing interaction strength $\left|U\right|/t$ in a uniform $50\times 50$ lattice with PBC for (a) $\alpha =1.5t,h=1.5t$ and (b) $\alpha =2t,h=1.5t$. The chemical potential without (with) the Hartree potential included is $\mu =-3t$ $\left(\tilde{\mu }=-3t\right)$ and $\mu =t$ $\left(\tilde{\mu }=t\right)$ for (a) and (b), respectively.

Figure 2

(a) Energy of the quasiparticle excitation bound to the nonmagnetic impurity scaled to the bulk spectral gap $E_0$ and (b) the real part of the order parameter at the impurity site in units of the bulk order parameter ${\mathrm{\Delta }}_0$ as a function of the inverse of the impurity potential, $t/V_{\mathrm{imp}}$, for $\mu =3.5t$ and $h=t$; for $\alpha =1.5t,t$, and $0.5t$, with the pairing interaction $U=-4.61t,-5.2t$, and $-6.24t$, respectively. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =1$.

Figure 3

(a) Real part of the order parameter in units of the bulk order parameter ${\mathrm{\Delta }}_0$ and (b) the average number of spin-up and spin-down electrons at the impurity site as a function of the impurity potential, $V_{\mathrm{imp}}/t$, for $\mu =3.5t,h=t,\alpha =t$ and $U=-5.2t$ in a $64\times 64$ lattice. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =1$.

Figure 4

Real [(a) and (c)] and imaginary [(b) and (d)] parts of the order parameter $\mathrm{\Delta }(x,y)/t$ as a function of $x$ and $y$ for $11\le x,y\le 41$ with $V_{\mathrm{imp}}=-50t$ at the center of a $51\times 51$ lattice for $\mu =3.5t$ and $h=t$; for $\alpha =t$ $\left(U=-5.2t\right)$ [(a) and (b)] and $\alpha =0.5t$ $\left(U=-6.24t\right)$ [(c) and (d)]. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =1$.

Figure 5

(a) Quasiparticle bound-state energy $E/E_0$ and (b) $\mathrm{Re}\left[\mathrm{\Delta }\left({\mathit{r}}_{\mathrm{imp}}\right)\right]/{\mathrm{\Delta }}_0$ as a function of $t/V_{\mathrm{imp}}$ for $\mu =-3t$ and $\alpha =1.5t$, for $h=1.5t,2t$, and $2.5t$ $(U=-5.1t,-6.02t$, and $-7.19t$, respectively) in a $51\times 51$ lattice. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =1$.

Figure 6

(a) Quasiparticle bound-state energy $E/E_0$ and (b) $\mathrm{Re}\left[\mathrm{\Delta }\left({\mathit{r}}_{\mathrm{imp}}\right)\right]/{\mathrm{\Delta }}_0$ as a function of $t/V_{\mathrm{imp}}$ for $\mu =-2t,h=2.5t,\alpha =2.5t$ and $U=-7t$ in a $51\times 51$ lattice. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =-1$.

Figure 7

Spin-up [(a) and (c)] and spin-down [(b) and (d)] components of the LDOS at the bound-state energy as a function of $x$ and $y$ for the entire $51\times 51$ lattice for $\mu =-2t,h=2.5t,\alpha =2.5t$, and $U=-7t$, with $V_{\mathrm{imp}}=-5t$ [(a) and (b)] and $V_{\mathrm{imp}}=6t$ [(c) and (d)] at the center of the lattice. ${\mathrm{\Delta }}_0\simeq 0.34t$ and $\nu =-1$.

Figure 8

Phase diagram of the topological phases with $\nu =0$ (trivial), $\nu =\pm 1$ (non-Abelian), and $\nu =-2$ (Abelian), separated by the phase boundaries, $h^2={(4t+\mu )}^2+{\mathrm{\Delta }}_0^2,h^2={\mu }^2+{\mathrm{\Delta }}_0^2$ and $h^2={(4t-\mu )}^2+{\mathrm{\Delta }}_0^2$ as a function of $h$ and $\mu$ with ${\mathrm{\Delta }}_0=0.34t$ [4]. Two values of the Zeeman field, $h=1.5t$ and $h=2.5t$, are indicated by dashed lines.

Figure 9

(a) Spin-up and (b) spin-down components of the LDOS as a function of excitation energy $E/t$ at lattice sites $(x,y)$ = (1,1), (2,1), (1,2), and (2,2) in the pure TCDW state for $\alpha =t,h=1.5t$, and $U=-4t$ at half filling in a $64\times 64$ lattice with PBC. ${\mathrm{\Delta }}_C\simeq 0.675t$ and $\nu =-2$.

Figure 10

(a) Spin-up and (b) spin-down electron density as a function of $x$ at $y=40$ and $y=41$ in the pure TCDW state for $\alpha =2.5t,h=1.5t$, and $U=-4t$ at half filling on a $80\times 80$ lattice with open boundaries at $x=1$ and $x=80$. ${\mathrm{\Delta }}_C\simeq 0.338t$ and $\nu =-2$.

Figure 11

Excitation spectrum in the pure TCDW state for $\alpha =2.5t,h=1.5t$, and $U=-4t$ at half filling on a $80\times 80$ lattice with open boundaries at $x=1$ and $x=80$. The index numbers the eigenvalues. ${\mathrm{\Delta }}_C\simeq 0.338t$ and $\nu =-2$.