Conductance in inhomogeneous quantum wires: Luttinger liquid predictions and quantum Monte Carlo results

We study electron and spin transport in interacting quantum wires contacted by noninteracting leads. We theoretically model the wire and junctions as an inhomogeneous chain where the parameters at the junction change on the scale of the lattice spacing. We study such systems analytically in the appropriate limits based on Luttinger liquid theory and compare the results to quantum Monte Carlo calculations of the conductances and local densities near the junction. We first consider an inhomogeneous spinless fermion model with a nearest-neighbor interaction and then generalize our results to a spinful model with an on-site Hubbard interaction.

Figure 1

A quantum wire connected to two noninteracting leads. The identical leads are modeled as a chain with hopping $t_{\ell }$, chemical potential ${\mu }_{\ell }$, and interaction $U_{\ell }=0$. The wire has parameters $U_w,t_w,{\mu }_w$. The junction between the leads and the wire is modeled as being abrupt.

Figure 2

The charge conductance $g_{x,y}^c\left({\omega }_n\right)$ for a noninteracting chain of spinful fermions with length $L=200$, periodic boundary conditions, and inverse temperature $\beta t=50$. Results for different distances $x-y$ between the perturbation and the measurement are displayed; for $x-y=0$, results are also shown for $\beta t=20$ (red circles). They all extrapolate in the $\omega \to 0$ limit to a conductance of $g_{x,y}^c(\omega \to 0)\to 2e^2/h$ as expected; see main text. The errors in the data are smaller than the point size.

Figure 3

The spin conductance $g_{x,y}^s\left({\omega }_n\right)$ for the same system and parameters as in Fig. 2. All curves extrapolate in the $\omega \to 0$ limit to the theoretically expected ideal conductance of $g_{x,y}^s(\omega \to 0)\to 0.5{\mu }_B^2/h$. As for Fig. 2, the errors are smaller than the point size.

Figure 4

Comparison between QMC data for the homogeneous Hubbard chain (2) with parameters $\mu =t,\beta t=50$, and $L=150$, and the exact Bethe ansatz results. (a) Charge conductance; (b) charge compressibility.

Figure 5

The Hubbard chain at half filling $\left(\mu =0,\beta t=50,L=150\right)$. The charge conductance only drops slowly to zero because of the exponentially small charge gap at small $U$. The drop in $g^c$ is well described by Eq. (8) (blue dashed line). The spin conductance, on the other hand, is independent of $U$ and fixed by the SU(2) symmetry (red dashed line).

Figure 6

QMC data for the charge and spin conductances of a junction of two noninteracting wires (symbols) with $t_w=t,\mu =1.5t,\beta t=50$, and $L=200$ compared to the exact result (lines).

Figure 7

Comparison between QMC data (symbols) for the inhomogeneous spinless chain (1) with parameters $\mu =0,t_w=t,U_{\ell }=0,\beta t=50,U_w=1.8t$ (black), $U_w=0.4t$ (blue), and $L=400$, and the analytical expression, given by Eq. (16) (solid red line and dashed purple line), with $\alpha$ a fitting parameter. The dotted line shows perfect conductance $\overline{K}{e}^2/h$, given by Eq. (15), with $\overline{K}\approx 0.739$ and $\overline{K}\approx 0.940$, respectively. The arrows at the $t_{\ell }$ axes indicate the points $u_{\ell }=u_w$.

Figure 8

Conductance $G-\overline{K}{e}^2/h$ for the inhomogeneous spinless chain (1) with parameters $t_w=t,U_w=1.8t,U_{\ell }=0,\beta t=50$, and $L=400$, for different chemical potentials $\mu$. The density varies between $n\approx 0.51$ $\left(\mu /t=0.1\right)$ and $n\approx 0.75$ $\left(\mu /t=1.5\right)$. The dashed line indicates perfect conductance. The solid lines show fits to $-A{(u_{\ell }-B)}^2$ with $A$ and $B$ fitting parameters [see Eq. (17)], the most general correction consistent with the model. The arrows indicate the points when $u_{\ell }=B$.

Figure 9

Charge and spin conductance for the inhomogeneous half-filled Hubbard model for a chain with $L=150,t_w=t,\beta t=25$, and $U_w/t=1$. We find ideal spin conductance for $t_{\ell }\approx 1$.

Figure 10

Relative differential charge conductance $g_{\lambda }^c=g^c-g_0^c$ for different values of $U_w$ as a function of $t_w$ in an inhomogeneous system of length $L=150$ with $t_{\ell }=t,\beta t=25$ and constant filling $n=1/4$ along the chain.

Figure 11

Relative differential spin conductance $g_{\lambda }^s=g^s-g_0^s$ for different values of $U_w$ as a function of $t_w$ in an inhomogeneous system of length $L=150$ with $t_{\ell }=t,\beta t=25$ and constant filling $n=1/4$ along the chain.

Figure 12

Local density at a lead-wire junction. Symbols denote the QMC results; the solid line denotes the Luttinger liquid result formula (31). The numerical data are obtained for an inhomogeneous Hubbard model with length $L=150$, hoppings $t_{\ell }=t,t_w=0.8t$, wire interaction $U_w=2t$, and inverse temperature $\beta t_w=25$. The chemical potentials are ${\mu }_{\ell }=1.7t,{\mu }_w=1.7t$. Error bars for the QMC data are smaller than the symbols.