Topological mirror insulators in one dimension

We demonstrate the existence of topological insulators in one dimension (1D) protected by mirror and time-reversal symmetries. They are characterized by a nontrivial ${\mathbb{Z}}_2$ topological invariant defined in terms of the “partial” polarizations, which we show to be quantized in the presence of a 1D mirror point. The topological invariant determines the generic presence or absence of integer boundary charges at the mirror-symmetric boundaries of the system. We check our findings against spin-orbit coupled Aubry-André-Harper models that can be realized, e.g., in cold-atomic Fermi gases loaded in one-dimensional optical lattices or in density- and Rashba spin-orbit-modulated semiconductor nanowires. In this setup, in-gap end-mode Kramers doublets appearing in the topologically nontrivial state effectively constitute a double-quantum dot with spin-orbit coupling.

Figure 1

Phase diagram and spectra of the spin-orbit coupled AAH model with $\alpha =\gamma =1/2$ (dimerization), $\beta =1/4,V_0=0, {\lambda }_0=0.5t_0,\delta \lambda =-0.3t_0,{\varphi }_t={\varphi }_{\lambda }=\pi$: (a) Half-filling phase diagram of the bulk for ${\varphi }_V=-\pi /4$. The value of the ${\mathbb{Z}}_2$ invariant $\nu$ is indicated by pixel color. Note that it only assumes two distinct values, 0 and 1. (b)–(d) Bulk band structures for $\delta V=0.5t_0$ and fixed $\delta t$ with (b) $\delta t=-0.1t_0$, (c) $\delta t=\delta t_c\approx 0.025$, and (d) $\delta t=0.1t_0$. The band structures correspond to systems along the arrow in (a). Note that the half-filling bulk energy gap closes away from the time-reversal invariant points $k=0$ and $\pi$.

Figure 2

Spectra of the spin-orbit coupled AAH model in a finite geometry for different ${\varphi }_V$. Other parameters are $\alpha =\gamma =1/2$ (dimerization), $\beta =1/4,V_0=0,\delta V=0.5t_0{\lambda }_0=0.5t_0,\delta \lambda =-0.3t_0,{\varphi }_t={\varphi }_{\lambda }=\pi$: (a) $\delta t=-0.5t_0$, (b) $\delta t=0.5t_0$. States localized at the ends of the chain are highlighted in red. In addition, we display the Hall conductivities $\sigma$ in units of $e^2/h$ associated with each gap when ${\varphi }_V$ is interpreted as an additional momentum variable.

Figure 3

Spectra and local charge densities of the mirror-symmetric, spin-orbit coupled AAH model with open boundary conditions and $\alpha =\gamma =1/2,\beta =1/4, V_0=0,\delta V=0.5t_0,{\lambda }_0=0.5t_0,\delta \lambda =-0.3t_0, {\varphi }_t={\varphi }_{\lambda }=\pi ,{\varphi }_V=-\pi /4$, and a surface potential $V_{\mathrm{LR}}=0.6t_0$: (a) Topological phase with $\delta t=-0.5t_0$, (b) trivial phase with $\delta t=0.4t_0$. The main panels show the energy spectra with end states highlighted in red. The dashed line signifies the chemical potential $\mu$ used for the calculation of the local charge densities ${\rho }_j$. The latter are presented in the insets. In addition, we display the corresponding values of the electronic end charges $Q_{\mathrm{L}}$ and $Q_{\mathrm{R}}$.

Figure 4

Effect of disorder in the half-filled, spinful AAH model with open boundary conditions and $\alpha =\gamma =1/2, \beta =1/4,t=-0.5t_0{V}_0=0,\delta V=0.5t_0,{\lambda }_0=0.5t_0, \delta \lambda =-0.3t_0,{\varphi }_t={\varphi }_{\lambda }=\pi$, and ${\varphi }_V=-\pi /4$ (topological phase): (a) Disorder-averaged IPR $\langle I_{\psi }\rangle$ for bulk states and end states. (b) Disorder-averaged end charges $\langle Q_{\mathrm{L}}\rangle$ and $\langle Q_{\mathrm{R}}\rangle$ with the chemical potential right below the in-gap end states. Shown is the dependence on the standard deviation $\gamma$ of the random disorder potential.

Figure 5

A semiconductor nanowire setup with perpendicular modulated voltage gates. The gates induce a modulation of both on-site potentials and Rashba SOC. Subject to mirror symmetry, such a nanowire hosts localized in-gap Kramers pairs at its ends.

Figure 6

Phase diagram and spectra of a density- and Rashba spin-orbit-modulated semiconductor nanowire with $\beta =\gamma =1/4, V_0=0.5t_0,{\lambda }_0=0.5t_0,\delta t=0,{\varphi }_V=3\pi /4$, and ${\varphi }_{\lambda }=\pi$: (a) Quarter-filling phase diagram. The value of the ${\mathbb{Z}}_2$ invariant $\nu$ is indicated by pixel color. The arrow shows a path through the phase diagram. (b) Energy spectra of nanowires with open boundary conditions at fixed $\delta \lambda =-0.4t_0$ corresponding to $\delta V$ values along the path shown in (a). States localized at the end points of the wires are highlighted in red. Note that there are end states at filling fractions $1/4, 1/2$, and $3/4$ in certain ranges of $\delta V$.