Conductance of fractional Luttinger liquids at finite temperatures

We study the electrical conductance in single-mode quantum wires with Rashba spin-orbit interaction subjected to externally applied magnetic fields in the regime in which the ratio of spin-orbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multi-electron interaction and applied magnetic field leads to a partial gap in the spectrum. We study how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire. We use two complementing techniques based on bosonization: refermionization of the model at a particular value of the interaction parameter and a semiclassical approach within a dilute soliton gas approximation of the functional integral. We show how the low-temperature fractional conductance can be affected by the finite length of the wire, by the properties of the contacts, and by a shift of the chemical potential, which takes the system away from the resonance condition. We also predict an internal resistivity caused by a dissipative coupling between gapped and gapless modes.

Figure 1

Single-band Rashba quantum wire with leads attached from the left and right side at positions $x=0$ and $x=L$, respectively. A magnetic field $B$ is applied along the wire axis in $x$ direction. Repulsive electron-electron interactions in the quantum wire are described within a Luttinger liquid model with interaction parameter $K_c<1$. The leads are assumed to be noninteracting with $K_c=1$.

Figure 2

(a) Spectrum of a Rashba quantum wire which consists of two spin-orbit split bands. A helical gap is opened by the Zeeman term at $k=0$ if a magnetic field is applied perpendicular to the SOI vector, which determines the quantization axis. A horizontal dashed line shows the position of the Fermi energy at filling factor $\nu =1/3$, at which the three-particle scattering (shown by black arrows) conserves momentum. Dotted arrows depict virtual transitions to/from states with a momentum close to zero. (b) Diagram for the multiparticle scattering process ($n=1$) corresponding to the interaction term given by Eq. (3). The right (left)-moving electrons with momentum close to the Fermi points $\pm 2k_{\mathrm{so}}/3,\pm 4k_{\mathrm{so}}/3$ are labeled as $R_s$ ($L_s$) with the spin label $s=\uparrow ,\downarrow$, whereas right (left)-moving electrons with momentum close to $k=0$ are labeled by $\tilde{R},\tilde{L}$. The interplay between electron-electron interactions at characteristic momenta $2k_F\equiv 2k_{\mathrm{so}}/3$ (wavy lines) and magnetic field $B$ (circle vertex) results in a partial gap at $\nu =1/3$.

Figure 3

Dependence of conductance $G$ at the filling factor $\nu ={\gamma }_c^{-1}=1/3$ on length $L$ at zero temperature ($T=0$) for the interaction parameter $K_c^{*}\approx 0.18$. The value of the gap $\mathrm{\Delta }$ renormalized by interactions is defined by Eqs. (7)–(11). The blue line shows the result obtained numerically (see Appendix pp2) when the tunneling contribution defined in Eq. (49) is taken into account; the dashed line shows the fractional conductance $G_{\nu =1/3}=e^2/5h$. The tunneling contribution to the conductance vanishes exponentially for long wires ($L\gg \hslash s/\mathrm{\Delta }$), $G\approx G_{\nu =1/3}+G_0{e}^{-2\mathrm{\Delta }L/s}$. The obtained conductance differs from the fractional value $G_{\nu =1/3}$ in short wires $L\lesssim \hslash s/\mathrm{\Delta }\sim \hslash v_F/\mathrm{\Delta }$ as the tunneling through a gap becomes significant. In the limit of a short wire, $L\ll \hslash s/\mathrm{\Delta }$, the fractional conductance is no longer observed, $G=2G_0=2e^2/h$. Note that in this limit the correlation length in Eq. (7) is determined by the length of the wire, $l_c\sim L$, and the gap is given by Eq. (11).

Figure 4

Effective transmission probability $\mathcal{T}$ [see Eq. (57)] for a long wire (blue curve) and short wires (red and green curves). Calculations have been performed numerically (see Appendix pp2) for an abrupt coordinate dependence of the gap, $\mathrm{\Delta }\left(x\right)=\mathrm{\Delta }\mathrm{\Theta }\left(x\right)\mathrm{\Theta }(L-x)$. At energies less than the gap $\mathrm{\Delta }$, the effective transmission is reduced due to a scattering at the effective potential barrier. At higher energies, Fabry-Perot oscillations emerge with a period $\delta \varepsilon \sim hs/2L$. The transmission becomes ideal, $\mathcal{T}=1$, in the limit $\varepsilon \gg \mathrm{\Delta }$. Note, that for the short wire (green curve) the gap has been calculated using Eq. (11).

Figure 5

Conductance $G$ of a quantum wire in the regime of fractional helical liquid, $\nu =1/3$, as function of temperature for different lengths of the wire: $L=5\hslash s/\mathrm{\Delta }$ (blue curve), $\hslash s/\mathrm{\Delta }$ (red curve), and $0.5\hslash s/\mathrm{\Delta }$ (green curve). Results are obtained numerically [see Eq. (58)] by using with effective transmission probability $\mathcal{T}$ from Fig. 4. Even at temperatures $T\gtrsim \mathrm{\Delta }$, the conductance does not reached its full value $2e^2/h$. We note that at high temperatures, $G$ only weakly depends on the length $L$.

Figure 6

Effective transmission probability $\mathcal{T}$ for different profiles of partial gap $\mathrm{\Delta }\left(x\right)$ [see Eq. (6)]: $l\ll l_{\mathrm{\Delta }}=\hslash s/\mathrm{\Delta }$ (blue curve), $l=0.5l_{\mathrm{\Delta }}$ (green curve), and $l=l_{\mathrm{\Delta }}$ (orange curve). In case of a smooth gap profile, the Fabry-Perot oscillations are washed out already at $\varepsilon \sim \mathrm{\Delta }$ while, in case of an abruptly changing gap, they vanish only at $\varepsilon \gg \mathrm{\Delta }$. The length of the wire is fixed to $L=5\hslash s/\mathrm{\Delta }$.

Figure 7

Conductance $G$ of a quantum wire of length $L=20\hslash v_F/L$ in the regime of fractional helical liquid, $\nu =1/3$, as function of temperature for different profiles of the gap given by Eq. (6): $l\gg \hslash v_F/\mathrm{\Delta }$ (red line), $l=\hslash v_F/\mathrm{\Delta }$ (orange line), $l=0.5\hslash v_F/\mathrm{\Delta }$ (green line), and $l\to 0$ (blue line). The results were obtained using Eq. (58) with effective transmission probability calculated numerically (see Fig. 6). The dependence shows a steeper activation behavior as the gap profile becomes smoother.

Figure 8

Schematic energy diagram for a smooth gate potential drop at the contact in two limiting cases: (a) small gate potential variation, $\delta V_g<{\mu }^{*}$; (b) large gate potential variation, $\delta V_g>{\mu }^{*}$. If the gate potential defined by Eq. (69) [solid blue line] lies close to the resonance value of the chemical potential, i.e., within the range of ${\mu }_1-{\mu }^{*}<V_g<{\mu }_1+{\mu }^{*}$ (shaded pink region), the measured conductance takes a fractional value, $G_{\nu =1/3}=e^2/5h$. The dotted blue lines show the corresponding bounds on the gate potential. In panel (a), the range of values of $V_g^{\mathrm{lead}}$ at which quantized values of conductance $G_{\nu }$ can be observed is given by $2{\mu }^{*}$. In contrast to that, in panel (b), this range is broader and can be estimated as ${\mu }^{*}+\delta V_g$.

Figure 9

Dependence of conductance $G$ on the shift of chemical potential $\delta \mu$ for different values of gate potential variation $\delta V_g$ at the contact at zero temperature $T=0$. The partial gap and gate potential profiles are defined by Eqs. (6) and (69) with $l=l_V=5\hslash s/\mathrm{\Delta }$ and the length of the wire is fixed to $L=20\hslash s/\mathrm{\Delta }$. The shift of chemical potential $\delta \mu$ measures the difference between the gate potential $V_g\left(x\right)$ in the bulk of the wire and at the resonance value of chemical potential ${\mu }_1$, see Fig. 8. At high values of gate voltage variation, $\delta V_g\gtrsim {\mu }^{*}\approx 0.3\mathrm{\Delta }$, the dip in conductance caused by opening of the partial gap broadens, see also Fig. 8.

Figure 10

The same as Fig. 9 but for an abrupt drop of the gate voltage at contacts, $l=l_V=0.1\hslash s/\mathrm{\Delta }$. If the shift of the chemical potential is less than ${\mu }^{*}\approx 0.3\mathrm{\Delta }$, the fractional conductance is observed. For higher values of the shift, the conductance is still suppressed in comparison to the quantized conductance $2G_0=2e^2/h$. This suppression is caused by a mismatch of modes in the leads and in the wire, similarly to the one observed in Ref. [48] for noninteracting systems. The reflection at the contacts gives rise to well-pronounced Fabry-Perot oscillations with a period of order of $\hslash s/L$.

Figure 11

Conductance of a long wire, $L\gg \hslash v_F/{\mathrm{\Delta }}_0$, as function of temperature $T$ at filling factor $\nu =1/3$ assuming that friction experienced by solitons can be disregarded [see Eq. (128)]. The temperature is given in units of the bare (nonrenormalized) gap ${\mathrm{\Delta }}_0\simeq B{U}_{2k_F}^2/v_F^2$, while the conductance is measured in units of $G_0=e^2/h$. At $T=0$ the conductance coincides with $G_{\nu }=e^2/5h$, while at high temperatures, $T\gg \mathrm{\Delta }$, the conductance reaches its full value $2e^2/h$. The activation curve becomes steeper as the interaction parameter $K_c$ approaches the critical value $K_c=1/3$. The activation temperature is determined by the renormalized value of the gap $\mathrm{\Delta }$ and is governed by Eq. (10). The length of the wire was taken much longer than the correlation length $\hslash v_F/\mathrm{\Delta }$.

Figure 12

Temperature dependence of the difference $\delta G=G-G_{\eta =0}$ between the total conductance $G$ at the filling factor $\nu =1/3$ with friction taken into account [see Eq. (129)] and the conductance $G_{\eta =0}$ calculated disregarding effects of friction [see Eq. (128)] for two values of Luttinger liquid parameter: (a) $K=0.1$ and (b) $K=0.2$. Temperature is measured in units of the bare gap ${\mathrm{\Delta }}_0\simeq B{U}_{2k_F}^2/v_F^2$, while the conductance is measured in units of $G_0=e^2/h$. The temperature dependence of the gap is determined using Eq. (10). The friction coefficient $\eta$ is estimated using Eq. (109). At high temperatures $T>\mathrm{\Delta }$, the conductance is suppressed by the friction. The correction to the conductance due to friction becomes more significant in case of strong interactions [see panel (a)]. The effect of friction may be neglected for weak interaction, if the length of the wire $L$ is not much longer than the correlation length $\hslash v_F/\mathrm{\Delta }$ [see panel (b)]. In contrast to that, if the wire is longer, the friction plays an important role also in this case.

Figure 13

The RG flow of the scaling dimension $D\left(l\right)$ of the sine-Gordon term in the action $S_1$ [see Eq. (14)] obtained by numerical solution of the RG equations [see Eqs. (A4, A5, A6, A7, A8)] for two different values of Luttinger liquid parameter $K_c$, which are chosen to be small enough so that the sine-Gordon term ${\omega }_0^{2}cos\left(\sqrt{2}\lambda {\varphi }_1\right)$ in Eq. (A1) generating the gap is relevant. As the flow parameter grows, the scaling dimension of the sine-Gordon term becomes small, $D\ll 1$, indicating strong pinning of the field ${\varphi }_1$. Although the initial scaling dimension is lower for a stronger interaction $K_c=0.1$, the scaling dimension at the point where the flow stops (shown with vertical arrows) is smaller for $K_c=0.2$. For initial values, we took $z={10}^{-3}$ and velocities for a corresponding value of interaction parameter $K_c$ from Table 1.

Figure 14

The same as Fig. 13 but for the flow of the interaction parameter $K_1\left(l\right)$. Although, the interaction parameter decreases during the flow, it does not change significantly.

Figure 15

The RG flow of velocities $v_1$ (solid lines) and $v_2$ (dashed lines) as a function of $v_{12}$. The initial values of Luttinger liquid parameter are chosen as $K_c=0.2$ (blue lines) and $K_c=0.1$ (red lines). All velocities are plotted in units of $v_F$ and vertical arrows show a point at which the flow is stopped. First, $v_2$ remains constant during the RG flow in accordance with RG equations [see Eq. (A8)]. Second, $v_1$ flows to a value $v_1^{*}$ close to $v_F$ (at $K_c=0.1$ and $K_c=0.2$ the corresponding values of $v_1^{*}$ are $v_1^{*}\approx 1.04v_F$ and $v_1^{*}\approx 1.2v_F$, respectively). Whereas, an amplitude of the cross-term $v_{12}$ in the action $S_{12}$ [see Eq. (A3)] flows to a much smaller value, $v_{12}<v_1$, which means that the cross term given by Eq. (A3) can be treated as a perturbation.