Helical Luttinger Liquid in Topological Insulator Nanowires

We derive and analyze the effective low-energy theory for interacting electrons in a cylindrical nanowire made of a strong topological insulator. Three different approaches provide a consistent picture for the band structure, where surface states forming inside the bulk gap correspond to one-dimensional bands indexed by total angular momentum. When a half-integer magnetic flux pierces the nanowire, we find a strongly correlated helical Luttinger liquid topologically protected against weak disorder. We describe how transport experiments can detect this state.

Figure 1

Band structure of a TI nanowire with $R=15\mathrm{nm}$ obtained by numerical diagonalization of Eq. (2). Points refer to bulk states, lines to surface states. Inset: Density $⟨\rho ⟩$ (dashed red line) and spin density [$⟨s_{\varphi }⟩$; blue solid line, $⟨s_z⟩$; black curve] vs radial coordinate for the right-moving state $(k,j)=(0.02{\AA }^{-1},1/2)$.

Figure 2

Same as Fig. 1 but using the tight-binding model on a diamond lattice, see Eq. (3). We take ${\widehat{e}}_z$ along the (111) axis, $R=5a$, ${\lambda }_{\mathrm{so}}=t$, and $\delta t=0.28t$. The size of the unit cell along this direction is $d_{\mathrm{cell}}=\sqrt{3}a$. Inset: as in Fig. 1 but for $k{d}_{\mathrm{cell}}=2.9$.

Figure 3

Numerical results for the lowest surface state gap ${\Delta }_s$ vs nanowire radius $R$, obtained from the low-energy approach (2) [black circles] and from the tight-binding model (3) [red squares]. The analytical prediction $v_2/R$, see Eq. (5), is given as a blue dashed curve. Inset: $\Phi$ dependence of ${\Delta }_s$ for $R=5a$ from the tight-binding approach, where the flux is introduced via Peierls phases.