Nonlinear Conductance of Long Quantum Wires at a Conductance Plateau Transition: Where Does the Voltage Drop?

We calculate the linear and nonlinear conductance of spinless fermions in clean, long quantum wires, where short-ranged interactions lead locally to equilibration. Close to the quantum phase transition, where the conductance jumps from zero to one conductance quantum, the conductance obtains a universal form governed by the ratios of temperature, bias voltage, and gate voltage. Asymptotic analytic results are compared to solutions of a Boltzmann equation which includes the effects of three-particle scattering. Surprisingly, we find that for long wires the voltage predominantly drops close to one end of the quantum wire due to a thermoelectric effect.

Figure 1

Electron density, $n(x)=n_0\mathbf{(}1+\delta n(x)\mathbf{)}$, at the QCP ($\mu =0$) calculated from a solution of the Boltzmann equation (1), using (a) a linearized collision integral for $L=100{\ell }_{\mathrm{eq}}$ and $V\to 0$ and (b) the nonlinearized collision integral for $L=10{\ell }_{\mathrm{eq}}$ and larger voltages, $eV/T=0.4$ and $eV/T=-0.6$ (inset: $eV/T=0.4$). The inset shows $\mu (x)$, $\delta T(x)=T(x)-T$, and $u(x)$ (in units of $T$ and $\sqrt{2T/m}$) obtained from fitting the local charge, energy, and momentum densities to Eq. (6). While in the linear response regime, $L\ll |{\ell }_V|$, there is a linear voltage drop across the wire, the voltage (and the density) drops predominantly close to one of the two contacts for $L\gg |{\ell }_V|$. Note that due to a finite drift $u$, the chemical potentials close to the contacts do not match the chemical potential $\mu \pm eV/2$ in the leads (shown as separate dots at $x=\pm L/2$ in the insets).

Figure 2

Conductance of fully equilibrated (solid line) and noninteracting electrons (dashed line) in the linear response regime (${\ell }_{\mathrm{eq}}\ll L\ll |{\ell }_V|$). Numerical results (symbols) agree with Eq. (13). Inset: upon lowering $T$, ${\ell }_{\mathrm{eq}}$ grows rapidly and a crossover from the equilibrated to the noninteracting conductance is observed for ${\ell }_{\mathrm{eq}}\sim L$ ($L=10{\ell }_{\mathrm{eq}}$, $\mu =0$).

Figure 3

Nonlinear conductance, $j_c/V$ obtained for $L=7.5{\ell }_{\mathrm{eq}}$ from a numerical solution of the Boltzmann equation. For $eV/T\to 0$ a linearized collision integral was used. Inset: within our numerical precision, there is no finite size dependence of the nonlinear conductance for $L\gg {\ell }_{\mathrm{eq}}$ ($L$ measured in units ${\ell }_{\mathrm{eq}}$).