Fractional Conductance in Strongly Interacting 1D Systems

We study one-dimensional clean systems with few channels and strong electron-electron interactions. We find that in several circumstances, even when time-reversal symmetry holds, they may lead to two-terminal fractional quantized conductance and fractional shot noise. The condition on the commensurability of the Fermi momenta of the different channels and the strength of the interactions resulting in such remarkable phenomena are explored using Abelian bosonization. Finite temperature and length effects are accounted for by a generalization of the Luther-Emery refermionization at specific values of the interaction strength, in the strongly interacting regime. We discuss the connection of our model to recent experiments in a confined two-dimensional electron gas, featuring possible fractional conductance plateaus, including situations with a zero magnetic field, when time-reversal symmetry is conserved. One of the most dominant observed fractions, with two-terminal conductance equal to $\frac{2}{5}(e^2/h)$, is found in several scenarios of our model. Finally, we discuss how at very small energy scales the conductance returns to an integer value and the role of disorder.

Figure 1

(a) Two-band dispersion with an example of a backscatering ${\mathcal{O}}_{\mathrm{bs}}$ process which conserves momentum when the chemical potential (horizontal dashed line) is such that $n_1{k}_1=n_2{k}_2$. (b) Similarly to (a), an umklapp process ${\mathcal{O}}_{\mathrm{um}}$ with a net momentum change conserves lattice momentum when $n_1{k}_1+n_2{k}_2=\pi (\text{integer})$. (c) An example of a time-reversal invariant backscattering process, $(n_1,n_2)=(3,1)$, occurring for fractional filling of a Rashba nanowire; see Supplemental Material, Sec. S.C [13]. (d) Illustration of a scenario where both bands interact throughout out the wire, yet one is confined and does not reach the reservoirs. This leads to a variety of possible fractional conductance values [13].

Figure 2

Conductance in a gate-voltage sweep for the time-reversal invariant nanowire described by the Hamiltonian density ${\mathcal{H}}_R(k)=(1/2m){(k+{\sigma }_z\alpha )}^2-\mu$, around the filling corresponding with $(n_1,n_2)=(1,3)$. (a) Plateaus at $(T/\mathrm{\Delta })=0.15$, varying interband separation (left to right), $(\alpha /\sqrt{2m\mathrm{\Delta }})=1$, 1.2, 1.5, 2, 2.5. (b) Plateaus at $(\alpha /\sqrt{2m\mathrm{\Delta }})=1.4$ and different temperatures (left to right), $(T/\mathrm{\Delta })=0.1$, 0.2, 0.3, 0.4, 0.5. Plots are shifted horizontally for clarity.

Figure 3

Schematic depiction of the conductance as a function of temperature. At temperatures above $T_L$ the behavior is determined by the ratio $\mathrm{\Delta }/T$, with an exponentially small correction to the universal fractional value $g$ at low temperatures, and a power-law behavior at high temperatures. Below $T_L$, the exponentially small energy scale $T_x$ determines the universal power law of the conductance.