Persistent current and Drude weight in mesoscopic rings

TL;DR

This study investigates how interactions and impurities affect persistent current and conductivity in mesoscopic rings, revealing phase-dependent decay behaviors and invariance properties using advanced numerical methods.

Contribution

It demonstrates the phase transition effects on persistent current decay and the invariance of current under impurity transformation in a spinless fermion system.

Findings

Persistent current decays algebraically in the LL phase and exponentially in the CDW phase.

Persistent current is invariant under impurity transformation $ ho o 1/ ho$ for large systems.

Drude weight vanishes as system size approaches infinity in both phases.

Abstract

We study the persistent current and the Drude weight of a system of spinless fermions, with repulsive interactions and a hopping impurity, on a mesoscopic ring pierced by a magnetic flux, using a Density Matrix Renormalization Group algorithm for complex fields. Both the Luttinger Liquid (LL) and the Charge Density Wave (CDW) phases of the system are considered. Under a Jordan-Wigner transformation, the system is equivalent to a spin-1/2 XXZ chain with a weakened exchange coupling. We find that the persistent current changes from an algebraic to an exponential decay with the system size, as the system crosses from the LL to the CDW phase with increasing interaction $U$. We also find that in the interacting system the persistent current is invariant under the impurity transformation $\rho\to 1/\rho $, for large system sizes, where $\rho $ is the defect strength. The persistent current exhibits a decay that is in agreement with the behavior obtained for the Drude weight. We find that in the LL phase the Drude weight decreases algebraically with the number of lattice sites $N$, due to the interplay of the electron interaction with the impurity, while in the CDW phase it decreases exponentially, defining a localization length which decreases with increasing interaction and impurity strength. Our results show that the impurity and the interactions always decrease the persistent current, and imply that the Drude weight vanishes in the limit $N\to \infty $, in both phases.