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

Abstract

The persistent current $(I)$ 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}⪡1$. The electrons interact via a screened interaction of range $d$. If $d\ensuremath{\lesssim}1/2{k}_{F}$, the interaction modifies the transmission as ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{N}^{\ensuremath{-}\ensuremath{\alpha}}$ and the current as $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $N$ is the electron number, $L$ the ring length, $N/L$ a fixed density, and $\ensuremath{\alpha}$ 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 ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ and $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$, both depending on two interaction parameters, $d$ and $\ensuremath{\alpha}$. To explain why our results depend solely on $\ensuremath{\alpha}$, we show analytically that the interaction-matrix elements depend only on $\ensuremath{\alpha}$ (not on $d$) when $d\ensuremath{\lesssim}1/2{k}_{F}$. As $d$ exceeds $1/2{k}_{F}$, our result for $LI/e{v}_{F}$ starts to depend on $d$ as well and likely approaches the result $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ in the limit $L⪢d⪢1/2{k}_{F}$, not feasible by our numerical methods.