Mobile-Electron Ising Ferromagnet

The general mobile-electron Ising ferromagnet (introduced by Fisher) is described, and its properties discussed: Electrons migrate over a lattice of spin-½ ions; when there are $k$ electrons on the bond ($i, j$), their energy is ${\epsilon }_k$ and they induce an Ising spin coupling between the spins $i$ and $j$. The mean number of electrons per bond $n$ is determined by the condition of electroneutrality. The model can be solved exactly in terms of the free energy of the underlying lattice and exhibits renormalized critical exponents: in the three-dimensional model, these are $\beta =0.33-0.36\mathrm{}$, $\gamma \simeq 1.43$, and ${\alpha }_s\simeq -0.14\mathrm{}$. The behavior of the simplest ferromagnetic model with $k=0, 1, or 2$ and $n=2\mathrm{}$ depends, for a fixed mode of electron-ion coupling, only on $n$ and $\epsilon =2\mathrm{}{\epsilon }_1-{\epsilon }_0-{\epsilon }_2$. The variation of the critical point with $n$, $\epsilon$, and the coupling energies is studied, and the behavior of the energy, magnetization, specific heat, and susceptibility is investigated. For certain parametric ranges, it is found that the spontaneous magnetization initially increases with $T$. The model exhibits a critical concentration for ordering ($n=r_c$), and also a lower critical temperature below which $M_0\mathrm{}\left(T\right)\mathrm{}$ again vanishes. A brief survey of related models exhibiting one or the other of these properties is presented.