Local magnetic moments due to loop currents in metals

Abstract

We present Hartree-Fock calculations on a simple model to obtain the conditions of formation of local magnetic moments due to loop currents ${L}_{o}$ and spin-loop currents ${L}_{s}$ and compare them to the conditions of formation of local spin moments $M$ which were given long ago in a similar approximation by Anderson. A model with three degenerate orbitals sitting on an equilateral triangle, with on-site and nearest-neighbor repulsions $U$ and $V$, respectively, and intersite kinetic energy, hybridizing with conduction electrons with a parameter $\mathrm{\ensuremath{\Delta}}$ is investigated. ${L}_{o}$ and ${L}_{s}$ are promoted by large $V/\mathrm{\ensuremath{\Delta}}$ and their magnitude is relatively unaffected by $U/\mathrm{\ensuremath{\Delta}}$. Spin-magnetic moments $M$ promoted by large $U/\mathrm{\ensuremath{\Delta}}$ on the other hand are adversely affected by $V/\mathrm{\ensuremath{\Delta}}$. In this model, ${L}_{o}$ for $V$ multiplied by the number of neighbors is approximately the same as the $M$ promoted by $U$ in Anderson's local model for $M$. ${L}_{o}$ and ${L}_{s}$ are degenerate if exchange interactions and Hund's rule are neglected but ${L}_{o}$ is favored when they are included. Many of the qualitative results are visible in an expression for the Hartree-Fock ground state energy derived as a function of small ${L}_{o}, {L}_{s}$, and $M$. Numerical minimization of the Hartree-Fock energy is presented for larger values. We also briefly discuss the connection and differences of the interaction-generated orbital currents and spin currents discussed here and generalized to a lattice with the topological states in metals and semiconductors.