Density-of-states picture and stability of ferromagnetism in the highly correlated Hubbard model

The problem of stability of saturated and nonsaturated ferromagnetism in the Hubbard model is considered in terms of the one-particle Green’s functions. Approximations by Edwards and Hertz and some versions of the self-consistent approximations based on the $1∕z$ expansion are considered. The account of longitudinal fluctuations turns out to be essential for description of the nonsaturated state. The corresponding pictures of density of states are obtained. “Kondo” density-of-states singularities owing to spin-flip processes are analyzed. The critical electron concentrations for instabilities of saturated ferromagnetism and paramagnetic state are calculated for various lattices. Drawbacks of various approximations are discussed. A comparison with the results of previous works is performed.

Figure 1

Density of states for the bare semielliptic DOS at concentration of carriers current $\delta =0.02$. (a) Lines 1 $\left(\sigma =\uparrow \right)$ and 2 $\left(\sigma =\downarrow \right)$ correspond to the non-self-consistent approximation (9); lines 1 $\left(\sigma =\uparrow \right)$ and 3 $\left(\sigma =\downarrow \right)$ to the non-self-consistent approximation with account of spin dynamics (20); (b) lines 4 $\left(\sigma =\uparrow \right)$ and 5 $\left(\sigma =\downarrow \right)$ correspond to the self-consistent approximation (17), and lines 4 $\left(\sigma =\uparrow \right)$ and 6 $\left(\sigma =\downarrow \right)$ to the Edwards-Hertz approximation (24).

Figure 2

Density of states for the bare semielliptic DOS at $\delta =0.20$. (a) Lines 1 $\left(\sigma =\uparrow \right)$ and 2 $\left(\sigma =\downarrow \right)$ correspond to the non-self-consistent approximation with spin dynamics (12); (b) lines 3 $\left(\sigma =\uparrow \right)$ and 4 $\left(\sigma =\downarrow \right)$ correspond to the self-consistent approximation (17), and lines 3 $\left(\sigma =\uparrow \right)$ and 5 $\left(\sigma =\downarrow \right)$ to the Edwards-Hertz approximation (24).

Figure 3

Density of states for the bare semielliptic DOS at $\delta =0.35$. (a) Lines 1 $\left(\sigma =\uparrow \right)$ and 2 $\left(\sigma =\downarrow \right)$ correspond to the non-self-consistent approximation with spin dynamics (12); (b) lines 3 $\left(\sigma =\uparrow \right)$ and 4 $\left(\sigma =\downarrow \right)$ correspond to the self-consistent approximation (17).

Figure 4

The dependence of the magnetization $⟨S^z⟩$ on $\delta$ for a number of bare DOS’s. Line 1 corresponds to rectangular DOS, line 2 to semielliptic DOS, lines 3, 4, 5, 6 to square, simple cubic (sc), bcc, and fcc latices, respectively.