Rigorous upper bound for the persistent current in systems with toroidal geometry

TL;DR

This paper establishes a rigorous upper bound on the persistent current in toroidal systems, valid across dimensions, interactions, impurities, and magnetic fields, providing a fundamental limit for such quantum phenomena.

Contribution

It introduces a universal upper bound for persistent currents in toroidal geometries, applicable in various dimensions and under diverse physical conditions.

Findings

The persistent current's absolute value is bounded by $e  N /4   m r_0^2$.

The bound holds for arbitrary interactions, impurities, and magnetic fields.

Valid in both two and three dimensions.

Abstract

It is shown that the absolute value of the persistent current in a system with toroidal geometry is rigorously less than or equal to $e \hbar N /4 \pi m r_0^2$, where $N$ is the number of electrons, and $r_0^{-2} = \langle r_i^{-2}\rangle$ is the equilibrium average of the inverse of the square of the distance of an electron from an axis threading the torus. This result is valid in three and two dimensions for arbitrary interactions, impurity potentials, and magnetic fields.