Band magnetism in the Hubbard model.

Abstract

A self-consistent moment method is applied to the Hubbard model in order to find out under what circumstances spontaneous band magnetism may occur. The theory is formulated for a two-sublattice structure to treat simultaneously para-, ferro-, and antiferromagnetic systems. The starting point is a two-pole ansatz for the one-electron spectral density, the free parameters of which are fitted by equating exactly calculated spectral moments. All correlation functions appearing in the moments can be expressed by the spectral density, guaranteeing therewith a closed system of equations, which can be solved self-consistently for the average particle numbers 〈${n}_{i\ensuremath{\uparrow}}$〉 and 〈${n}_{i\ensuremath{\downarrow}}$〉. A T=0 phase diagram is presented in terms of band occupation n (0\ensuremath{\le}n\ensuremath{\le}2) and Coulomb interaction U. Ferromagnetic solutions appear only if n exceeds a critical occupation ${n}_{c}^{\mathrm{FM}}$ and U a minimum value ${U}_{\mathrm{min}}$. For antiferromagnetic solutions a critical U does not exist, but a critical band occupation ${n}_{c}^{\mathrm{AFM}}$ does.Antiferromagnetism is stable in a restricted region of n around n=1, which is broadest for intermediate couplings (U/W\ensuremath{\simeq}1, W being the Bloch bandwidth) and shrinks to the n=1 axis for strong couplings (U/W\ensuremath{\rightarrow}\ensuremath{\infty}). For smaller n, but n>${n}_{c}^{\mathrm{FM}}$ and sufficiently high U (U>W), ferromagnetism is stable, while for low band occupations the system is paramagnetic irrespective of U. The critical temperatures ${T}_{C}$ and ${T}_{N}$ are strongly U and n dependent. For fixed n, ${T}_{C}$ increases with U, but saturates for U\ensuremath{\rightarrow}\ensuremath{\infty} at finite values (500--800 K), while ${T}_{N}$ has a maximum at an intermediate U value (U\ensuremath{\simeq}W). First-order as well as second-order transitions are observed. Ferromagnetic order arises mainly because of a shift of \ensuremath{\uparrow} and \ensuremath{\downarrow} quasiparticle subbands. In antiferromagnets, corresponding \ensuremath{\uparrow} and \ensuremath{\downarrow} subbands occupy exactly the same energy regions, but with different state densities. The magnetic behavior of the Hubbard model can be understood as a direct consequence of the sensitive (T,n,U) dependence of the quasiparticle density of states, which is therefore discussed in detail.