Three-dimensional fractional topological insulators in coupled Rashba layers

We propose a model of three-dimensional topological insulators consisting of weakly coupled electron- and hole-gas layers with Rashba spin-orbit interaction stacked along a given axis. We show that in the presence of strong electron-electron interactions the system realizes a fractional strong topological insulator, where the rotational symmetry and condensation energy arguments still allow us to treat the problem as quasi-one-dimensional with bosonization techniques. We also show that if Rashba and Dresselhaus spin-orbit interaction terms are equally strong, by doping the system with magnetic impurities, one can bring it into the Weyl semimetal phase.

Figure 1

Schematic representation of the system formed by tunnel-coupled layers with charge carriers. The unit cell consists of two electron (blue) and two hole (green) layers. The color brightness encodes the two different signs of the SOI.

Figure 2

Dispersion relation of the layers for a fixed value of $\theta$. (a) The chemical potentials $\mu$ (black lines) are tuned to the SOI energy $E_{so}$. The colors blue/green encode positive/negative helicity. The arrows represent the tunneling processes between fields allowed by spin and momentum conservation laws. (b) The chemical potential is tuned to $E_{so}/9$. In the presence of strong interactions, tunneling processes assisted by backscattering dominate resulting in the bulk gap. The orange and black arrows represent terms in ${\mathcal{O}}_{t_1}$ and ${\mathcal{O}}_{t_2}$, respectively [see Eqs. (12) and (13)].

Figure 3

The spectrum in the topological phase obtained numerically for $N_y=300, t_1/t=0.2, t_2/t=0.1$, and $\overline{\alpha }/t=0.3$; see Appendix pp1. The bulk states (blue) are fully gapped with gap ${\mathrm{\Delta }}_{\text{min}}=2(t_1-t_2)$. The dispersion of the surface states localized in the $xz$ plane (green) is represented by an anisotropic Dirac cone. The inset shows the helical spin structure in the layer $\eta =\overline{\tau }=1$ at $y/a_y=1$ for $E/t=0.02$ (dashed line), confirming the presence of a single helical Dirac cone at the $xz$ surface.

Figure 4

The probability density of the wave function $|{\psi }_{1,\overline{1},\uparrow }{|}^2$ in the ($1,\overline{1}$) layer for a state on the Dirac cone on the first one-hundred sites ($N_y=800$) along the $y$ direction. The figure was obtained for $t_1/t=0.2, t_2/t=0.1, \overline{\alpha }/t=0.3$, and $\mu =-4t$ for a state on the Dirac cone with $k_x{a}_x/\pi =0.05$ and $k_z{a}_z/\pi =0$. The wave function is localized at the $xz$ surface decaying rapidly along $y$ direction into the bulk.

Figure 5

The spectrum in the trivial phase obtained numerically for $N_y=300, t_1/t=0.1, t_2/t=0.2$, and $\overline{\alpha }/t=0.3$. The spectrum is fully gapped and no surface states were found in the $xz$ plane. The same holds for all other surfaces, which shows that here the system is in the trivial phase.

Figure 6

The spectrum of the topological phase in the presence of disorder along the cut $k_z{a}_z/\pi =0$ with parameters $N_y=300, t_1/t=0.2, t_2/t=0.1, \overline{\alpha }/t=0.3, \mu =-4t$, and $\sqrt{\langle \delta {\mu }^2\rangle }=0.7t_2$. The two Dirac cones on two opposite surfaces (green and orange) still exist and there is no gap opened by disorder. However, due to slightly different disorder configurations at two surfaces, there is a small shift of the position of the center of the Dirac cone in energy.

Figure 7

Schematics of a backscattering assisted tunneling process between two neighboring electron layers. The two Fermi surfaces are drawn on top of each other. For brevity, the spin polarizations of the corresponding state at the Fermi surface for the first (green arrows) and second layers (blue arrows) are indicated only for $k_x<0$ ($k_x>0$). The involved electrons residing in the respective layers are shown by blue (green) dots. The momentum transfer $\vec{q}$ (orange dotted arrows) during the tunneling event is compensated by the two backscattering processes such that $\vec{q}+{\vec{q}}_1+{\vec{q}}_2=0$. Correspondingly, due to the spin structure of the Fermi surface, the amplitude of the tunneling process with momentum transfer $\vec{q}$ connecting two states with misaligned spins is reduced. The backscattering assisted tunneling amplitude (and thus the resulting bulk gap) is maximum if all involved spins are aligned, thus, $\vec{q}\parallel {\vec{q}}_{1,2}$. In this case, the rotational symmetry of the system is preserved.

Figure 8

The Weyl semimetal spectrum for the parameters $t_1/E_{so}=0.3, t_2/E_{so}=0.19$, and $k_z=0$. Left: Four Weyl nodes exist for a weak exchange interaction ($J/E_{so}=0.08$). Right: In the regime $|J-t_1|<t_2$, only two Weyl outer nodes remain ($J/E_{so}=0.17$).

Figure 9

Spin structure of the states on the Weyl cones at an energy of $E/t=0.02$ at $y/a_y=260$ ($N_y=400$). For clarity we projected the spin onto the $k_x{k}_z$ plane. The parameters used for the numerics are $t_1/t=0.2, t_2/t=0.1, \overline{\alpha }/t=0.4$, and $J/t=0.07$.

Figure 10

Spectrum in the Weyl semimetal phase in half of the Brillouin zone (BZ) as a function of momenta ($k_x, k_z$). The spectrum of a semi-infinite system ($y>0$) was obtained numerically [cf. Eq. (A5)] for $N_y=800, \overline{\alpha }/t=0.45, t_1/t=0.3, t_2/t=0.19$, and $J/t=0.08$, i.e., in the regime $J<t_1-t_2$ where we expect four Weyl nodes from the analytical analysis. The bulk (surface) states are colored in blue (green). The spectrum is cut in half along the line connecting the gap closing points. The spectrum indeed features two (since we show only one half) gapless points as well as the Fermi arcs.