Topological states and interplay between spin-orbit and Zeeman interactions in a spinful Su-Schrieffer-Heeger nanowire

The interplay between the spin-orbit and Zeeman interactions acting on a spinful Su-Schrieffer-Heeger model is studied based on an InAs nanowire subjected to a periodic gate potential along the axial direction. It is shown that a nontrivial topological phase can be achieved by regulating the confining-potential configuration. In the absence of the Zeeman field, we prove that the topology of the chain is not affected by the Rashba spin-orbit interaction due to the persisting chiral symmetry. The energies of the edge modes can be manipulated by varying the magnitude and direction of the external magnetic field. Remarkably, the joint effect of the two spin-related interactions leads to novel edge states that appear in the gap formed by the anticrossing of the bands of a finite spinful dimerized chain, and can be merged into the bulk states by tilting the magnetic-field direction.

Figure 1

(a) A semiconductor wire subjected to gate potentials $V_1$ and $V_2$ arranged periodically along the axial direction $x$, to form a double quantum-dot chain. The length of the unit cell is $2L$, and $2d$ is the width of each potential wall. (b) The local potentials $V_{\mathrm{qd},1}\left(x\right)$ and $V_{\mathrm{qd},2}\left(x\right)$ that define two quantum dots (QD1 and QD2) within each unit cell, and the lowest orbital on each dot. Our analysis also applies for the case $V_1>V_2$ (not shown).

Figure 2

[(a) and (b)] The energy spectra of the spinless InAs double quantum-dot chain versus the wave vector $k$ for different potential differences $\mathrm{\Delta }V$. (c) The Bloch spectrum of the spinful InAs chain at zero Zeeman field ($B=0$), for $\mathrm{\Delta }V=5.0$ meV and $x_{\mathrm{so}}^{}=180$ nm. (d) The same as (c), for $B=0.2$ T tilted by $\theta =0.5\pi$ in the $y-z$ plane. The dashed (red) curves depict the energy levels of the Bloch spectrum, calculated from the continuous Hamiltonian (1), and the solid (blue) curves are derived from the discrete Hamiltonian (3).

Figure 3

(a) The energy spectrum of a finite spinless chain versus $\mathrm{\Delta }V$. The spatial density distributions of one spinless edge mode, $P_n$, for different values of $\mathrm{\Delta }V$, as indicated by the symbols in panel (a), are displayed in (b). (c) The energy spectrum of a finite chain of 20 sites versus ${\mathrm{\Delta }}_z^{}$ for $\mathrm{\Delta }V=10$ meV, $\theta =0.5\pi$, and zero spin-orbit coupling. The emergence of the Zeeman-split edge states is indicated by the thick (red) lines. (d) The Bloch spectrum of the semiconductor chain as a function of $k$ with ${\mathrm{\Delta }}_z^{}=t_{-}^{}, \mathrm{\Delta }V=10$ meV, $\theta =0.5\pi$, and zero spin-orbit coupling. Besides, the energy values of the spin-up and spin-down edge states are indicated by the (yellow) dash-dotted and (purple) dashed lines, respectively.

Figure 4

(a) The energy spectrum of a finite spinful InAs chain versus $\mathrm{\Delta }V$ for $B=0.2$ T and $\theta =\pi /2$ and in the absence of the SOI. The edge states appearing in the band gap are indicated by the thick (red) lines. (b) The same as in (a) but in the presence of the SOI. The coefficient ${\eta }_{20}^{}$ characterizes the degree of confinement of the edge states in a 20 site chain.

Figure 5

(a) The positive energy spectrum of a finite spin-orbit-active chain of 20 sites versus the Zeeman splitting, with $\mathrm{\Delta }V=10$ meV and $\theta =0.5\pi$. (b) The positive energy spectrum of the same chain versus the tilting angle $\theta$ of $\mathbf{B}$ in the $y-z$ plane for ${\mathrm{\Delta }}_z^{}=0.44$ meV and $\mathrm{\Delta }V=10$ meV. Also shown are the (yellow) shadowed regions where there exist four pairs of edge modes whose degree of confinement is characterized by the coefficient ${\eta }_{20}^{}$. (c) The spin-density distributions, $S_n^{}$, of the two edge modes marked by a square and a diamond in panel (a). Moreover, for a discrete chain comprising of 100 sites, the spin-density distribution of the new edge mode [corresponding to the diamond in panel (a)] is further exemplified in the inset. (d) The Bloch spectrum of the spin-orbit-active chain versus $k$ for $\mathrm{\Delta }V=10$ meV, ${\mathrm{\Delta }}_z^{}=0.44$ meV, and $\theta =0$. In this case, the anticrossing gap accommodating the spin-mixed edge states completely disappears, as indicated by the red circles. In addition, the inset shows the dependence of the two spin-flip tunneling amplitudes (in meV units) on $\theta$, with the (purple) solid and (yellow) dashed curves representing the inter- and intracell tunneling amplitudes, $|t_{\mathrm{ex}}^{\prime }|$ and $|t_{\mathrm{in}}^{\prime }|$, respectively.

Figure 6

(a) The spatial distribution of the confining potential $V_{\mathrm{DQD}-\mathrm{in}}^{}\left(x\right)$ that determines the intracell tunneling amplitude within the double quantum-dot chain in the main text (see Fig. 1). (b) The confining potential $V_{\mathrm{DQD}-\mathrm{ex}}^{}\left(x\right)$ that determines the intercell tunneling amplitude. (c) The intra-/intercell hopping amplitude of an InAs double quantum dot chain, $t_{\mathrm{in}/\mathrm{ex},0}^{}$, as a function of the potential difference $\mathrm{\Delta }V=V_1^{}-V_2^{}$, with the periodic length $2L=120$ nm and $d=10$ nm. Here $t_{\mathrm{in},0}^{}$ ($t_{\mathrm{ex},0}^{}$) is denoted by the solid (dashed) line and the variations of the chemical potential $V_c$ are displayed by the dash-dotted curve. (d) The intra- and intercell spin-flip tunneling amplitudes, $t_{\mathrm{in}}^{\prime }$ and $t_{\mathrm{ex}}^{\prime }$, of the double-quantum-dot chain as functions of the tilting angle $\theta$, with the potential difference $\mathrm{\Delta }{V}_{}^{}=-10$ mev, the magnetic field strength $B=1.0$ T, and the spin-orbit interaction scaled by $x_{\mathrm{so}}^{}\equiv \hslash /\left(m_e\alpha \right)=180$ nm.

Figure 7

The spatial distribution of the confinement potential $U\left(x\right)$ in a unit cell; the separation of the unit cell into four regions (denoted by I, II, III, and IV) in which the Bloch functions are derived (see text).