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

摘要

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 t_K_F « 1. The electrons interact via a screened interaction of range d. If d≤1/2K_F, the interaction modifies the transmission as t_K_F≈t_K_FN~(-α) and the current as LI/ev_F≈t_K_FN~(-α), where N is the electron number, L the ring length, NIL a fixed density, and α 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≈t_K_F(L/d)~(-α) and LI/ev_F ≈|t_K_F|(L/d)~(-α), both depending on two interaction parameters, d and a. To explain why our results depend solely on a, we show analytically that the interaction-matrix elements depend only on α (not on d) when d ≤ 1/2K_F. As d exceeds 1/2K_F, our result for Ll/ev_F starts to depend on d as well and likely approaches the result Ll/ev_F≈|t_K_F|(L/d_~(-α) in the limit L»d»1/2k_F, not feasible by our numerical methods.