Tenfold d‐Electron Degeneracy in the Hubbard Model and Transition‐Metal Magnetism. II. The Antiferromagnetic‐Paramagnetic and Antiferromagnetic‐Ferromagnetic Transitions

Abstract Intrasite d‐d electron interactions are included in the Hubbard model, replacing the simplified s‐electron picture of Penn [5]. The technique is identical with an earlier paper by Siegel and Kemeny [1] on the ferromagnetic‐paramagnetic transition. Choosing k‐space correlation functions to represent the paramagnetic, ferromagnetic, and antiferromagnetic states, the tenfold degenerate eigenvalue problem is solved and free energies are self consistently calculated for various combinations of the three parameters: d‐band filling, direct interaction Coulomb constant divided by one third the d‐bandwidth, and exchange interaction constant divided by one third the d‐bandwidth. The antiferromagnetic‐paramagnetic and antiferromagnetic‐ferromagnetic phase boundary surfaces are plotted a t near zero and finite temperatures with the third dimension being the exchange interaction constant divided by one third the d‐bandwidth (J/4T). The near zero temperature trend shows that the inclusion of exchange, as expected, enhanced the antiferromagnetic region at the expense of the paramagnetic region. This antiferromagnetic region enhancement over the para‐magnetic region is exactly the opposite of the nondegenerate. t‐matrix result of Caron and Kemeny [7]. Finite temperature calculations for the antiferromagnetic‐paramagnetic transition show the same trend with increasing exchange, but shift the phase boundary so as to decrease the antiferromagnetic region relative to that at near zero temperature for the same value of exchange interaction constant, J. The antiferromagnetic‐ferromagnetic phase boundary shift as a function of increasing d‐d exchange interaction constant, J, is much harder to calculate since two ordered magnetic spin states are in competition. Within the scatter of the calculated free energy values, it is seen that the ferromagnetic region is enhanced at the expense of the antiferromagnetic region, but just barely. This enhancement is much less than that of ferromagnetic over paramagnetic. in [1], or of antiferro‐magnetic over paramagnetic, here, where strongly spin ordered magnetic spin states are compared to a spin disordered state. For the slight ferromagnetic region enhancement over the antiferromagnetic region, again the trend is opposite to the t‐matrix result, which enhances the antiferromagnetic region at the expense of the ferromagnetic region relative to Penn's [5] nondegenerate calculation. But again, since Penn's [5] calculation is good only to direct Coulomb interaction constant K = 0.25, while Caron and Kemeny's [7] is good rigorously down to K = 0.0, nothing can be said abont, how the calculation compares with Penn's [5] for K < 0.25.