Thermodynamics of itinerant magnets in a classical spin-fluctuation model

Thermodynamics of itinerant magnets is studied using a classical model with one parameter characterizing the degree of itinerancy. Monte Carlo simulations for bcc and fcc lattices are compared with the mean-field approximation and with the Onsager cavity field approximation extended to itinerant systems. The qualitative features of thermodynamics are similar to the known results of the functional integral method. It is found that magnetic short-range order is weak and almost independent on the degree of itinerancy, and the mean-field approximation describes the thermodynamics reasonably well. Ambiguity of the phase space measure for classical models is emphasized. The Onsager cavity field method is extended to itinerant systems, which involves the renormalization of both the Weiss field and the on-site exchange interaction. The predictions of this approximation are in excellent agreement with Monte Carlo results.

Figure 1

(Color online) [(a) and (b)] Reduced magnetization $⟨x_z⟩$, [(c) and (d)] mean squared local moment $⟨x^2⟩$, and [(e) and (f)] inverse paramagnetic susceptibility ${\chi }^{-1}$ as a function of the reduced temperature $t=T/\left(J_0{m}_0^2\right)$. MFA results are shown by solid (blue online) lines for $g\left(x\right)=1$ and by dashed black lines for $g\left(x\right)=x^{-2}$. MC results are displayed by black circles for $g\left(x\right)=1$ and by red (gray) squares for $g\left(x\right)=x^{-2}$ (in both cases the symbols are filled for fcc and empty for bcc lattice). The inset in panel (e) highlights the region close to $t_c$ for the bcc lattice with $g\left(x\right)=1$ and also shows the results of the generalized Onsager method (black line connecting the MC points).

Figure 2

(Color online) (a) Reduced Curie temperature $t_c$ and (b) MSRO parameter $⟨\text{cos}{\theta }_{nn}⟩$ at $T=1.1T_c$ as a function of the itinerancy parameter $\alpha$ for the bcc lattice. Solid black line, red (gray) squares, and blue (dark gray) circles show the results of MFA, MC, and the generalized Onsager method for $g\left(x\right)=1$, respectively. Dashed black line and empty black squares depict MFA and MC results for $g\left(x\right)=x^{-2}$. Green (light gray) triangles represent the incomplete Onsager reaction field correction with the on-site interaction left unrenormalized. The blue (gray) dashed-dotted line in the upper panel shows the effective moment $x_{\text{eff}}^2$ found from the Curie constant for $g\left(x\right)=1$ in MFA. Very similar results were obtained for the fcc lattice (not shown).