IV) Materials calculations with dynamical mean field theory (DMFT)

Ab-initio electronic Hamiltonian(non-relativistic/Born-Oppenheimer approximation)kinetic energy lattice potential Coulomb interactionH = X i"− 2 ∆ i2m e+ X l−e 24πɛ 0Z l|r i − R l |#+ 1 2Xi≠je 24πɛ 01|r i − r j |LDA bandstructure corresponds toH LDA = X i"− 2 ∆ i2m e+ X l−e 24πɛ 01|r i − R l | + Zd 3 r#e2 1LDAρ(r) + Vxc (ρ(r i ))4πɛ 0 |r i − r|LDA + local Coulomb interactionĤ = X ɛ LDAklm ĉ†klσĉkmσ+ 1 XU σσ′lm2 ˆn ilσ ˆn imσ ′ − ∆ɛ X ˆn imσU’ U’−Jklmσi lσmσ| {z }′ imσm=1H LDA U• non-local Coulomb interaction → Hartree term (to leading order in 1/Z; contained in LDA)• constrained LDA ⇒ U, J, V =U−2J, ∆ɛ(McMahan et al.’88, Gunnarsson et al.’89)m=2