Variational Approach to Finite-temperature Magnetism in the Degenerate Narrow Bands

The variational approach which describes the local electron correlations at finite temperatures has been extended to the degenerate-band case. It reduces Tc in the static approximation by a factor of two, gives realistic Tc between the observed value Tc (Obs) and 2Tc (Obs) , and suppresses the temperature independent charge fluctuation in Fe and Ni. The finite-temperature theories of magnetism which interpolate between the weak and strong in&' = ~ l ln / [ ! (f) ' x e-pE(c~TI (2) teraction limits have extensively been applied to the narrow band systems to understand the local vs. itinerant behavior of magnetism [I]. Among them the Here P denotes the inverse temperature, and 3 = variational approach (VA) developed by Kakehashi U/2D $J (1 -b 1/20] , D being the degeneracy of the and Fulde [2] reduces to the Gutzwiller-type energy bands. Fe~nmann inequality to the free energy (2) is at the ground state, therefore takes into account the then written as local electron correlations missing in the static approxF < _ F t + ( E ( < , T ) E t ( E , T ) ) t . (3) imation (SA) which has been adopted in the most of other theories. Recent numerical investigations to the simple-cubic Hubbard model with half-filled band [3] have shown that the calculated ground-state energy and sublattice magnetization quantitatively agree with those in the variational Monte-Carlo method [4] in which the ground state quantities are exactly obtained for a given Gutzwiller wave function. Furthermore %he NBel temperatures calculated from the VA are expected to be much better than those obtained from the 4 x 4 x 4 Monte-Carlo simulation bv Hirsch [5]. Thus we believe that among various theories the VA describes best the finite temperature magnetism in the intermediate regime. So far the VA is limited to the single-band Hubbard model, thus it was impossible to argue the correlation effects at finite temperatures in realistic systems. We present in this paper an extension of the VA to the degenerate bands and clarify the role of the local electron correlations in Fe and Ni. The Hamiltonian which we adopt here is the degenerate-band Hubbard model with the following electron-electron interactions Here U and J are Coulomb and exchange energy parameters respectively, ni and Si denote the total chargeand spin-density operators on site i. According to the functional integral method the free energy F is exactly expressed by means of an energy functiond E (<, T) projected onto the longitudinal field variables {ti). Here Ft is the trial free energy with the trial energy functional Et (6, T) , and the average (. . .), means a classical average with respect to Et (E, T) . Since the characteristic temperature associated with the correlated motion of electrons is much higher than the Curie temperature Tc, most of the local electron correlations persist even above Tc. We therefore take them adiabatically into account assuming the following trial energy functional. Here Est (c, T) is the energy functional to the SA which reduces to the Hartree-Fock energy at T = 0. The second term is the Gutzwiller-type correlation energy [6] in the random exchange fields {ti}. H = H ( H ) , , and (. . .), means the average with respect to the independent electrons in the random exchange fields {ti). The operator Q ( r ] ) is defined by