Persistent currents in one dimension: the other side of Leggett's theorem

TL;DR

This paper investigates the sign and behavior of persistent currents in one-dimensional electron rings, establishing bounds and revealing conditions for diamagnetic, paramagnetic, or superconductor-like responses based on electron polarization and number.

Contribution

It introduces a topology-based method to derive lower bounds on free energy, complementing Leggett's upper bounds, and characterizes current behavior for various electron configurations.

Findings

Odd electron rings are diamagnetic.

Even electron rings are paramagnetic.

Unpolarized electrons with N multiple of four show superconductor-like behavior.

Abstract

We discuss the sign of the persistent current of N electrons in one dimensional rings. Using a topology argument, we establish lower bounds for the free energy in the presence of arbitrary electron-electron interactions and external potentials. Those bounds are the counterparts of upper bounds derived by Leggett. Rings with odd (even) numbers of polarized electrons are always diamagnetic (paramagnetic). We show that unpolarized electrons with N being a multiple of four exhibit either paramagnetic behavior or a superconductor-like current-phase relation.