Localized End States in Density Modulated Quantum Wires and Rings

We study finite quantum wires and rings in the presence of a charge-density wave gap induced by a periodic modulation of the chemical potential. We show that the Tamm-Shockley bound states emerging at the ends of the wire are stable against weak disorder and interactions, for discrete open chains and for continuum systems. The low-energy physics can be mapped onto the Jackiw-Rebbi equations describing massive Dirac fermions and bound end states. We treat interactions via the continuum model and show that they increase the charge gap and further localize the end states. The electrons placed in the two localized states on the opposite ends of the wire can interact via exchange interactions and this setup can be used as a double quantum dot hosting spin qubits. The existence of these states could be experimentally detected through the presence of an unusual $4\pi$ Aharonov-Bohm periodicity in the spectrum and persistent current as a function of the external flux.

Figure 1

The figure shows a quantum wire (placed horizontally) of length $L$ with negatively (upper vertical gates) and positively (lower vertical gates) charged gates forming a superlattice potential. Because of the induced charge-density modulation a bound state at each wire end can emerge.

Figure 2

(a) The part of the spectra around the gap of the Hamiltonian given by Eq. (1), obtained by exact diagonalization. The red bars denote two almost degenerate bound (midgap) states. We have chosen for the parameters $t=7$, $\Delta =0.8$, and $N=320$. (b) One of two bound states. Plotted here is ${\psi }_{+}={\psi }_L+{\psi }_R$, where ${\psi }_{L,R}$ are states localized at the left (right) end of the wire.

Figure 3

The dependence of the root-mean-square value $\sqrt{\sigma [E_i]}$ of the $i$th energy level ($i=1\dots N$, i.e., for all energy levels) on the standard deviation of the random disorder potential.

Figure 4

(a) Quantum wire in an Aharonov-Bohm ring geometry with negatively charged gates (on the top of the wire). The bound states are localized on either side of the weak link (grey) of strength $t_0$. The energy (b) and persistent current $j=-\partial F/\partial \Phi$ (c) dependence on the flux $\Phi /{\Phi }_0$ are plotted with the solid curves for the effective model [Eq. (7)] and with the dashed curves for the lattice model [Eq. (1)]. The parameters for the $2\pi$ periodic red and blue solid curves correspond to $\delta /t_0=2.2$ and $({ϵ}_{+}+{ϵ}_{-})/t_0=6.2$, and for the $4\pi$ periodic green curve to $\delta /t_0=2.0$ and $({ϵ}_{+}+{ϵ}_{-})/t_0=6.0$ [see Eq. (8)], while for the dashed curves the parameters are $\Delta /t_0=3.85$ ($2\pi$ periodic red and blue curves) and $\Delta /t_0=4.54$ ($4\pi$ periodic green curve). The ratio $\Delta /t=0.5$ and $N=50$ is the same for all three dashed curves. Here, $t_0$ is chosen such that we have a degeneracy at $\Phi /{\Phi }_0=2\pi$. Assuming only the lower bound state is filled, the persistent current shows an unusual $4\pi$ periodicity as a function of $\Phi /{\Phi }_0$.