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We present an \textit{ab-initio} disordered local moment theory for the Gibbsfree energy of a magnetic material. Two central objects are calculated: thelattice Fourier transform of the direct local moment - local moment correlationfunctions in the paramagnetic state and local internal magnetic fields asfunctions of magnetic order. We identify the potentially most stable magneticphases from the first, which can include non-collinear and long-period statesin complex multi-atom unit cells, and extract higher order correlations amongthe local moments from the second. We propose that these latter entitiesproduce a picture of effective multi-site magnetic interactions depending onthe state and extent of magnetic order and discuss its relation to otherapproaches. We show how magnetic phase diagrams for temperature, magneticfield, and lattice structure and also magnetocaloric and mechanocaloric effectscan be obtained from this approach. The theory accurately predicts the order oftransitions and quantifies contributions to first-order and order-ordermagnetic phase transitions from both purely electronic sources andmagnetoelastic effects. Our case study is the apparently frustrated magnetismof the Mn$_3$A class of materials in all its cubic, hexagonal, and tetragonalstructures. The theory produces magnetic phases and transition temperatures ingood agreement with experiment. We explain the first-order triangularantiferromagnetic to collinear antiferromagnetic transition in cubic Mn$_3$Ptas a magnetovolume driven effect, and its absence for A=Ir and Rh. We alsoconstruct the magnetic phase diagram of Mn$_3$Pt and explore its potential as abarocaloric material. Finally, we prepare the groundwork for future fullyrelativistic studies of the temperature dependence of the magnetism of Mn$_3$A,including Mn$_3$Sn, Mn$_3$Ga, and Mn$_3$Ge.

physical review. b./physical review. b

Ab initio theory of the Gibbs free energy and a hierarchy of local  moment correlation functions in itinerant electron systems: The magnetism of  Mn$_3$A materials class