Low-energy properties of fractional helical Luttinger liquids

We investigate the low-energy properties of (quasi) helical and fractional helical Luttinger liquids. In particular, we calculate the Drude peak of the optical conductivity, the density of states, as well as charge transport properties of the interacting system with and without attached Fermi liquid leads at small and large (compared to the gap) frequencies. For fractional wires, we find that the low-energy tunneling density of states vanishes. The conductance of a fractional helical Luttinger liquid is noninteger. It is independent of the Luttinger parameters in the wire, despite the intricate mixing of charge and spin degrees of freedom, and only depends on the relative locking of charge and spin degrees of freedom.

Figure 1

Helical wire attached to leads from the left and right sides at positions $x=\pm \frac{L}{2}$.

Figure 2

Real part of the conductivity $\text{Re}\left\{\sigma (0,0,\omega )\right\}$ as defined in Eq. (32) at frequencies much smaller than the gap. The solid line is the numerically evaluated conductivity for $\Delta =10000\sqrt{v_F}/L$, $K_c=0.5$, $K_s=1$, and $v_i=v_F/K_i$. Panel (a) shows data for $\delta =1$; panel (b) corresponds to $\delta =1/3$. The superimposed circles have been calculated from the known analytical expression [34, 35, 36] of $\sigma (0,0,\omega )$ for a spinless Luttinger liquid (LL) with effective Luttinger parameter ${\tilde{K}}_{\mathrm{eff}}$ and effective velocity $v_{\mathrm{eff}}$ coupled to spinless Luttinger liquid leads of velocity $v_F$ and Luttinger parameter $K_{\mathrm{eff},L}$, while the horizontal dashed lines indicate the values between which this latter analytical expression oscillates; see main text.

Figure 3

Real part of the conductivity $\text{Re}\left\{\sigma (0,0,\omega )\right\}$ at frequencies much larger than the gap. The solid line corresponds to $\Delta =0.01\sqrt{v_F}/L$, $K_c=0.5$, $K_s=1$, $v_i=v_F/K_i$, and $\delta =1$. The circles stem from the known expression [34, 35, 36] of $\sigma (0,0,\omega )$ for a spinful Luttinger liquid (LL) with the same parameters coupled to (here noninteracting) Luttinger liquid leads. Like in Fig. 2, the horizontal dashed lines indicate the values between which this latter expression oscillates. For $\omega \to 0$, the conductivity reaches the gapped value $1e^2/h$ in a sharp drop. Different values of $\delta$ lead to the same curve, up to a modified zero-frequency dip that recovers the conductance given in Eq. (31).

Figure 4

Real part of the conductivity $\text{Re}\left\{\sigma (0,0,\omega )\right\}$, evaluated numerically for $\Delta =130\sqrt{v_F}/L$, $K_c=0.5$, $K_s=1$, $v_i=v_F/K_i$, and $\delta =1$ (this value of $\Delta$ corresponds to a gap of ${\omega }_{\mathrm{gap}}\approx 184v_F/L$). At small frequencies, the behavior of a spinless Luttinger liquid is recovered, while the conductivity approaches the expression of a spinful and gapless Luttinger liquid at large frequencies. For frequencies close to the gap, not only is the expansion of the sine-Gordon potential unjustified, but the conductivity also shows some unphysical, noisy features (sections drawn with a gray dashed line).