Conducting fixed points for inhomogeneous quantum wires: A conformally invariant boundary theory

Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is “healing,” which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions.

Figure 1

The full density including Friedel oscillations near the boundary, numerical results (filled circles) are fitted to the analytical result of Eq. (3.19) (lines) with $t_l=1.308t_r$, $U_{\ell }=0$, and $U_r=1.8t_r$. The local potential energy is $V=-0.25t_r$ and $t_r\beta =10$. Underneath a schematic of the system under consideration is shown.

Figure 2

The full density including Friedel oscillations near the boundary, numerical results (filled circles) are fitted to the analytical result of Eq. (3.19) (lines) with $t_l=1.31t_r$, $U_{\ell }=0$, and $U_r=1.8t_r$. The local potential energy is $V=-0.75t_r$ and $t_r\beta =10$.

Figure 3

Plotted are the Friedel oscillations for different local potential energies $V$ calculated by QMC simulations, see main text for details, on the right-hand side of the junction ($x>0$). We only show the longer wavelength amplitude of the rapid oscillations. In each panel from top to bottom: $t_{\ell }=1.3t_r$ for (black) circles, $t_{\ell }=1.4t_r$ for (red) squares, $t_{\ell }=1.5t_r$ for (green) diamonds, $t_{\ell }=1.6t_r$ for (blue) up-triangles, and $t_{\ell }=1.7t_r$ for (purple) down-triangles. We have used everywhere $U_{\ell }=0$, $U_r=1.8t_r$ and inverse temperature $t_r\beta =10$.

Figure 4

The scaling of the local spin correlation functions $C_z\left(\tau \right)$ and $C_{\pm }\left(\tau \right)$ at the fixed point: $t_{\ell }=1.518t_r$, $U_{\ell }=0$, and $U_r=1.8t_r$. The magnetic field is zero (i.e., $V=0$ for the corresponding fermion system) and the temperature is $t_r\beta =25$. (a) Numerical data, black circles, are compared to the predicted scaling $f{\left(\tau \right)}^{\nu }$ with $f\left(\tau \right)\equiv {|sin(\pi \tau /\beta )|}^{-2}$ and $\nu =\overline{g}$ (red curve). As a comparison we also plot $f{\left(\tau \right)}^{\nu }$ with $\nu =g_{\ell }$, $g_r$, $\overline{\gamma }$, see Eq. (4.20). (b) Numerical data, black circles, are compared to the predicted scaling $f{\left(\tau \right)}^{1/4\nu }$ with $\nu =\overline{\gamma }$ (red curve). As a comparison we also plot $f{\left(\tau \right)}^{1/4\nu }$ with $\nu =g_{\ell }$, $g_r$, $\overline{g}$, see Eq. (4.19).

Figure 5

The scaling of the local spin correlation function $C_z\left(\tau \right)$ at the conducting fixed point with $t_{\ell }=1.518t_r$, $U_{\ell }=0$, and $U_r=1.8t_r$. The magnetic field is zero (i.e., $V=0$ for the corresponding fermion system). Numerical data for different inverse temperatures (as indicated on the plot) are compared to the predicted scaling (lines). As temperature is lowered the field theory becomes more accurate.

Figure 6

The scaling of the local spin correlation function $C_z\left(\tau \right)$ away from the conducting fixed point with $t_{\ell }=t_r$, $U_{\ell }=0$, and $U_r=1.8t_r$. The magnetic field is zero (i.e., $V=0$ for the corresponding fermion system). From top to bottom we plot different values of the inverse temperature $t_r\beta =\{0.5,2.5,5,10,15,20,25\}$.