Local magnetic moments due to loop currents in metals

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 inter-site kinetic energy, hybridizing with conduction electrons with a parameter ${\Delta}$ is investigated. $L_o$ and $L_s$ are promoted by large $V/{\Delta}$ and their magnitude is relatively unaffected by $U/{\Delta}$. Spin-magnetic moments $M$ promoted by large $U/{\Delta}$ on the other hand are adversely affected by $V/{\Delta}$. 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 semi-conductors.