| Summary: | We present an ab initio theory to describe magnetic ordering and magnetic phase transitions at finite temperatures from pairwise and multi-spin interactions. Our formalism is designed to model thermal fluctuations of disordered local moments associated with atomic sites and adequately describes how these emerge from the glue of many interacting electrons. The key ingredient is to assume a time-scale separation between the evolution of the local moment orientations and a rapidly responsive electronic background setting them. This is the Disordered Local Moment picture grounding the framework of our theory. The method uses Density Functional Theory calculations constrained to specific local moment configurations to model the electronic structure and exploits Green's functions within a Multiple Scattering Theory to solve the Kohn-Sham equations. Two central objects are calculated as functions of magnetic ordering: internal magnetic fields sustaining the local moments and the lattice Fourier transform of the interactions in the paramagnetic state. We develop a methodology to extract the pairwise and multi-spin constants from the first and use the second to study the magnetic interactions in the reciprocal space and gain information of the type and extent of most stable magnetic order. These quantities are directly related to the first and second derivatives of the free energy of a magnetic material, respectively. Hence, our approach is able to provide thermodynamic quantities of interest, such as temperature and entropy changes for the evaluation of caloric effects, and magnetic phase diagrams for temperature, magnetic field, and lattice spacing studies can be constructed. Transition temperatures and their order, as well as tricritical points, are obtainable. We apply the theory to carry out major investigations on long-period magnetic phases in the heavy rare earth elements (HREs) and magnetic frustration in the Mn-based antiperovskite nitride Mn3GaN. The mixing of both pairwise and four-site magnetic interactions have been found to have profound consequences on the magnetism of both systems. We have obtained a generic HRE magnetic phase diagram which is consequent on the response of the common valence electronic structure to the f-electron magnetic moment ordering. We also present a modelling based on the lanthanide contraction to describe ferromagnetic, helical antiferromagnetic, and fan phases in Gd, Tb, Dy, and Ho, in excellent agreement with experiment. Our study of Mn3GaN shows that its first-order paramagnetic-antiferromagnetic triangular transition originates from the fourth order terms and that the effect of biaxial strain to distort the compensated antiferromagnetic interactions has a large impact on the frustrated magnetism. As a consequence, new collinear magnetic phases stable at high temperatures are predicted and a very rich temperature-strain phase diagram is obtained. We also show how to get the best refrigerating performance and design a novel elastocaloric cooling cycle from the features of the diagram.
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