Junctions of one-dimensional quantum wires: Correlation effects in transport

We investigate transport of spinless fermions through a single site dot junction of $M$ one-dimensional quantum wires. The semi-infinite wires are described by a tight-binding model. Each wire consists of two parts, the noninteracting leads and a region of finite extent in which the fermions interact via a nearest-neighbor interaction. The functional renormalization group method is used to determine the flow of the linear conductance as a function of a low-energy cutoff for a wide range of parameters. Several fixed points are identified and their stability is analyzed. We determine the scaling exponents governing the low-energy physics close to the fixed points. Some of our results can already be derived using the non-self-consistent Hartree-Fock approximation.

Figure 1

A single site dot junction of $M$ quantum wires. Across the bonds of the lattice sites $j=1,\dots ,N$ (small filled circles) the fermions interact via a nearest-neighbor interaction, while they are noninteracting in the leads with $j>N$ (solid line). The hopping amplitude between the first site of wire $\nu =1,\dots ,M$ and the dot site is $t_{\nu }$. The on-site energy on the dot site (large filled circle) is $V$.

Figure 2

Effective exponents of $[4∕{(M-1)}^2]-{\mid t_{\nu ,{\nu }^{\prime }}\mid }^2$ (circles) and ${\mid t_{\nu ,M}\mid }^2$ (stars) as a function of ${\delta }_N$ for $M=5$, $\tilde{t}=1$, $t_M={10}^{-3}$, and different values of $U=-1,-0.5,0.5,1$ from top to bottom. On the scale of the plot the results obtained from $[4∕{(M-1)}^2]-{\mid t_{\nu ,{\nu }^{\prime }}\mid }^2$ and ${\mid t_{\nu ,M}\mid }^2$ are indistinguishable. The scaling exponent is read-off when a plateau is reached for ${\delta }_N<5\times {10}^{-4}$, which roughly corresponds to $N>{10}^4$.

Figure 3

Scaling exponent of the conductance for a weak link as a function of $M-1$ (symbols) for different $U$. Note the reciprocal scale of the $x$ axis. The lines show ${\gamma }_1\left(M\right)={\beta }_s∕(M-1)$.

Figure 4

One-parameter scaling plot of the effective transmissions ${\mid t_{\nu ,M}\mid }^2$ (lower curve) and ${\mid t_{\nu ,{\nu }^{\prime }}\mid }^2$ with $\nu ,{\nu }^{\prime }⩽M-1$ (upper curve) for $M=4$, $U=-1$, and $\tilde{t}=1$. The variable is ${\delta }_N∕{\delta }_0\left(t_M\right)$, with a nonuniversal scale ${\delta }_0\left(t_M\right)$. The asymptotic values of the transmission $4∕{(M-1)}^2$ and 0 reached for ${\delta }_N∕{\delta }_0\left(t_M\right)\to \infty$ as well as $4∕M^2$ reached for ${\delta }_N∕{\delta }_0\left(t_M\right)\to 0$ are indicated on the $y$ axis.

Figure 5

Scaling exponent of the conductance for a slightly modified link as a function of $M$ for different $U$. Note the reciprocal scale of the $x$-axis. The lines show the fit Eq. (56).

Figure 6

Flow of $g$ Eq. (59) for the case of a symmetric junction with on-site energy $V=0.5,1,1.5,2$, different $\tilde{t}∊[0.01,0.7]$, $M=2,3,10$ legs, and $U=-1$. The arrows indicate the direction of the flow for $U<0$. For $U>0$ it is reversed.

Figure 7

Scaling exponent of the conductance for a small on-site energy as a function of $M$ (symbols) for different $U$. Note the reciprocal scale of the $x$ axis. The lines show the fit Eq. (60).