Luttinger liquid and persistent current in a continuous mesoscopic ring with a weak link

The persistent current $\left(I\right)$ of the spinless-electron Luttinger liquid is calculated by numerical microscopic methods in the continuous one-dimensional ring containing a weak link with transmission amplitude ${\tilde{t}}_{k_F}⪡1$. The electrons interact via a screened interaction of range $d$. If $d\lesssim 1/2k_F$, the interaction modifies the transmission as $t_{k_F}\simeq {\tilde{t}}_{k_F}{N}^{-\alpha }$ and the current as $LI/e{v}_F\simeq \left|{\tilde{t}}_{k_F}\right|N^{-\alpha }$, where $N$ is the electron number, $L$ the ring length, $N/L$ a fixed density, and $\alpha$ depends on the interaction in accord with the renormalization-group (RG) theory. Unlike our results, the RG theory predicts for a finite $d$ the power laws $t_{k_F}\simeq {\tilde{t}}_{k_F}{\left(L/d\right)}^{-\alpha }$ and $LI/e{v}_F\simeq \left|{\tilde{t}}_{k_F}\right|{\left(L/d\right)}^{-\alpha }$, both depending on two interaction parameters, $d$ and $\alpha$. To explain why our results depend solely on $\alpha$, we show analytically that the interaction-matrix elements depend only on $\alpha$ (not on $d$) when $d\lesssim 1/2k_F$. As $d$ exceeds $1/2k_F$, our result for $LI/e{v}_F$ starts to depend on $d$ as well and likely approaches the result $LI/e{v}_F\simeq \left|{\tilde{t}}_{k_F}\right|{\left(L/d\right)}^{-\alpha }$ in the limit $L⪢d⪢1/2k_F$, not feasible by our numerical methods.

Figure 1

Persistent current $LI/e{v}_F$ versus $L$ and $N$ for $\varphi =0.25{\varphi }_0$. The transmission of the weak link is ${\left|{\tilde{t}}_{k_F}\right|}^2=0.03$. The top panel shows the BBCI results for various $\left(V_0,d,{\alpha }_{\text{RG}}\right)$ listed in Table . The numerical result for noninteracting electrons (dotted-dashed line) approaches at large $L$ the value $0.5\left|{\tilde{t}}_{k_F}\right|$, predicted for $\varphi =0.25{\varphi }_0$ by formula (1). The dashed lines show dependence (11), where $\alpha ={\left(1+2{\alpha }_{\text{RG}}\right)}^{1/2}-1$, ${\alpha }_{\text{RG}}$ is given by formula (15), and $C=1+1.66\alpha$. The BBCI data reach this asymptotic dependence at large $N$. Inset in the top panel shows the asymptotic BBCI data plotted as $2LI/e{v}_{F}C$ versus $x\equiv N^{-\alpha }$. The full line in inset is the function $f\left(x\right)=x\left|{\tilde{t}}_{k_F}\right|/\left|{\tilde{r}}_{k_F}\right|$. In the bottom panel we compare the selected BBCI data from the top panel with the FCI data. Inset in the bottom panel shows that the data follow the dependence $\propto \text{sin}\left(2\pi \varphi /{\varphi }_0\right)$, drawn in the full line. The dotted lines are a guide for the eyes.

Figure 2

Persistent current $LI/e{v}_F$ versus $L$ and $N$ for $\varphi =0.25{\varphi }_0$. We compare the CI results for two different transmissions ${\left|{\tilde{t}}_{k_F}\right|}^2$ and for various interactions with $\left(V_0,d\right)$ obeying the equation ${\alpha }_{\text{RG}}\left(V_0,d\right)=0.2565$. The dashed lines show formula (11). Inset shows the CI data plotted as the ratio $\Omega$ (see the text), where $I\left({\tilde{t}}_{k_F,1}\right)$ and $I\left({\tilde{t}}_{k_F,2}\right)$ are the persistent currents for ${\left|{\tilde{t}}_{k_F,1}\right|}^2=0.03$ and ${\left|{\tilde{t}}_{k_F,2}\right|}^2=0.003$.

Figure 3

Persistent current $LI/e{v}_F$ versus $L$ and $N$ for various $\left(V_0,d\right)$ chosen so that ${\alpha }_{\text{RG}}\left(V_0,d\right)=0.2565$ and $\alpha ={\left(1+2{\alpha }_{RG}\right)}^{1/2}-1=0.23$. The CI data are compared with Eqs. (11, 6). Equation (11) agrees with the CI data for $d\le 3\text{nm}\simeq 1/2k_F$ but fails to fit the CI data for $d>1/2k_F$ ($d=12$ and 24 nm) as they depend on $d$. To illustrate clearly how the dependence on $d$ disappears for small $d$, the inset shows again the BBCI data for $N=12$ and $N=38$ but in dependence on $d$. Equation (6) does not fit the CI data (see the text for details).