Nonlinear spin and orbital susceptibilities of a Mott insulator

We investigate the coupling of electromagnetic fields to electrons on a lattice with strong on-site Hubbard interactions and calculate the spin and orbital nonlinear electric field susceptibilities. Gauge covariance of the equations of motion for both the photon and matter fields prescribes a minimal coupling procedure in which the fermionic quantum momentum operator is boosted by the electromagnetic vector potential. On a lattice this minimal coupling procedure takes the form of a Peierls substitution whereby the electronic wavefunction acquires a phase upon transport through the electromagnetic field. Strong electron-electron interactions can localize the charge degrees of freedom on a lattice forcing the Peierls phase to vanish. We show how the relativistic corrections in Dirac equation give rise to effective spin-spin interactions that couple directly to an external electric field even in the absence of a Peierls phase.