Renormalization group study of transport through a superconducting junction of multiple one-dimensional quantum wires

We investigate transport properties of a superconducting junction of many $(N⩾2)$ one-dimensional quantum wires. We include the effect of electron-electron interaction within the one-dimensional quantum wire using a weak interaction renormalization group procedure. Due to the proximity effect, transport across the junction occurs via direct tunneling as well as via the crossed Andreev channel. We find that the fixed point structure of this system is far more rich than the fixed point structure of a normal metal–superconductor junction $(N=1)$, where we only have two fixed points—the fully insulating fixed point or the Andreev fixed point. Even a two-wire $(N=2)$ system with a superconducting junction, i.e., a normal metal–superconductor–normal metal structure, has nontrivial fixed points with intermediate transmissions and reflections. We also include electron-electron interaction induced backscattering in the quantum wires in our study and hence obtain non-Luttinger liquid behavior. It is interesting to note that (a) effects due to inclusion of electron-electron interaction induced backscattering in the wire, and (b) competition between the charge transport via the electron and hole channels across the junction give rise to a nonmonotonic behavior of conductance as a function of temperature. We also find that transport across the junction depends on two independent interaction parameters. The first one is due to the usual correlations coming from Friedel oscillations for spin-full electrons giving rise to the well-known interaction parameter $[\alpha =(g_2-2g_1)∕2\pi \hslash v_F]$. The second one arises due to the scattering induced by the proximity of the superconductor and is given by $[{\alpha }^{\prime }=(g_2+g_1)∕2\pi \hslash v_F]$. The nonmonotonic conductance and the identification of this new interaction parameter are two of our main results. In both the expressions $g_1=V\left(2k_F\right)$ and $g_2=V\left(0\right)$, where $V\left(k\right)$ is the interelectron interaction potential.

Figure 1

(Color online) Multiple wires connected to a superconducting junction. The dashed lines represent the fact that the model can be trivially extended to more than two wires. $a$ is the effective length of the superconductor.

Figure 2

(Color online) The processes that contribute to the amplitude for an incoming electron to transform to an outgoing hole. Note that all the processes shown here are to first order in the interaction parameters since they only involve a single scattering from a Friedel oscillation or the pair potential. Process (c) involves scattering from a pair potential before the electron reaches the junction. The remaining processes involve two reflections from the junction and a scattering from the Friedel oscillation or the pair potential. In the diagrams, $\xi =\frac{1}{2}\alpha r^{\star }\ln \left(kd\right)$, $\eta =-\frac{1}{2}{\alpha }^{\prime }{r}_A^{\star }\ln \left(kd\right)$, and ${\eta }^{\prime }=\frac{1}{2}{\alpha }^{\prime }{r}_A\ln \left(kd\right)$.

Figure 3

(Color online) Electron CT with bare amplitude $t$ is shown in (a) and CAR with bare amplitude $t_A$ is shown in (b).

Figure 4

(Color online) The extra processes that contribute to the amplitude for an incoming electron to transform to an outgoing hole on the same wire, due to the second wire. Processes (a) and (b) are transmitted to the second wire and reflected by the Friedel oscillation, whereas (c) and (d) are transmitted to the second wire and reflected by the pair potential.

Figure 5

(Color online) Schematic representation of the situation where a three-wire junction is hooked to the stable fixed point, AGFP. An incident electron in one wire is either reflected back as a hole in the same wire or is transmitted as a hole in another wire along with the addition of the two electrons into the superconductor forming a Cooper pair. The direction of RG flow from two unstable fixed points to the stable fixed point (AGFP) is also depicted on the bottom left side of the diagram.

Figure 6

(Color online) Conductance of the NS junction is plotted in units of $e^2∕h$ as a function of the dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$.

Figure 7

(Color online) Conductance $G_{\mathit{CA}}$ of the NSN junction is plotted (when the two leads have antiparallel spins) in units of $e^2∕h$ as a function of the dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$. The inset shows the behavior of the same conductance for fixed values of $\alpha$.

Figure 8

(Color online) Conductance $G_{\mathit{CT}}$ of the NSN junction (when the two leads have parallel spins) in units of $e^2∕h$ as a function of the dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$. The inset shows the behavior of the same conductance for fixed values of $\alpha$.

Figure 9

(Color online) (a): Conductance of the NSN junction $G_{\mathit{NSN}}={\mid t_A\mid }^2-{\mid t\mid }^2$ is plotted in units of $2e^2∕h$ as a function of the dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$. (b): Charge conductance $G_{\mathit{FSN}}={\mid t_A\mid }^2-{\mid t\mid }^2$ is plotted for the FSN case as a function of the dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature in units of $e^2∕h$ and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$.

Figure 10

(Color online) ${\mid t_A\mid }^2$ is plotted as a function of dimensionless parameter $l$ where $l=\ln (L∕d)$ and $L$ is either $L_T=\hslash v_F∕k_{B}T$ at zero bias or $L_V=\hslash v_F∕eV$ at zero temperature and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$. The set of curves in the top represents the RG flow of ${\mid t_A\mid }^2$ when the starting point is in the vicinity of $r_A=0$ fixed point, while the set of curves in the bottom half represents the RG flow of ${\mid t_A\mid }^2$ with starting point close to $r_A=-1$.