Dzyaloshinskii Interaction and Exchange-Relativistic Effects in Orthoferrites

We present an overview of the microscopic theory Dzyaloshinskii-Moriya (DM) coupling and related exchange-relativistic effects such as exchange anisotropy, electron-nuclear antisymmetric supertransferred hyperfine interactions, magnetogyrotropic effects, magnetoelectric in strongly correlated 3$d$ compounds focusing on orthoferrites RFeO$_3$ (R is a rare-earth ion or yttrium Y). % Particular emphasis will be placed role crystal field sense direction Dzyaloshinskii vector, its dependence superexchange bonding geometry, other subtle features effects. Most attention paper centers around derivation value, orientation, (sign) under different types (super)exchange interaction field. Microscopically derived expression for vector geometry allows one to find all overt hidden canting angles well corresponding contribution magnetic anisotropy. Being based theoretical predictions regarding sign we have predicted study detail novel phenomenon, {\it weak ferrimagnetism} mixed ferromagnets with competing signs vectors. The ligand NMR measurements are shown effective tool inspect DM external Along ferrimagnets RFe$_{1-x}$Cr$_x$O$_3$, although lesser extent, address typical $\alpha$-Fe$_2$O$_3$, FeBO$_3$, FeF$_3$.