Stoner criterion

The Stoner criterion is a condition to be fulfilled for the ferromagnetic order to arise in a simplified model of a solid. It is named after Edmund Clifton Stoner.

Stoner model of ferromagnetism


Ferromagnetism ultimately stems from Pauli exclusion. The simplified model of a solid which is nowadays usually called the Stoner model, can be formulated in terms of dispersion relations for spin up and spin down electrons,



E_\uparrow(k)=\epsilon(k)-I\frac{N_\uparrow-N_\downarrow}{N},\qquad E_\downarrow(k)=\epsilon(k)+I\frac{N_\uparrow-N_\downarrow}{N}, $$

where the second term accounts for the exchange energy, $$I$$ is the Stoner parameter, $$N_\uparrow/N$$ ($$N_\downarrow/N$$) is the dimensionless density of spin up (down) electrons and $$\epsilon(k)$$ is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If $$N_\uparrow +N_\downarrow$$ is fixed, $$E_\uparrow(k), E_\downarrow(k)$$ can be used to calculate the total energy of the system as a function of its polarization $$P=(N_\uparrow-N_\downarrow)/N$$. If the lowest total energy is found for $$P=0$$, the system prefers to remain paramagnetic but for larger values of $$I$$, polarized ground states occur. It can be shown that for



ID(E_{\rm F}) > 1 $$

the $$P=0$$ state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the $$P=0$$ density of states at the Fermi energy $$D(E_{\rm F})$$.

A non-zero $$P$$ state may be favoured over $$P=0$$ even before the Stoner criterion is fulfilled.

Relationship to the Hubbard model
The Stoner model can be obtained from the Hubbard model by applying the mean-field approximation. The particle density operators are written as their mean value $$\langle n_i\rangle$$ plus fluctuation $$n_i-\langle n_i\rangle$$ and the product of spin-up and spin-down fluctuations is neglected. We obtain



H = U \sum_i [n_{i,\uparrow} \langle n_{i,\downarrow}\rangle +n_{i,\downarrow} \langle n_{i,\uparrow}\rangle - \langle n_{i,\uparrow}\rangle \langle n_{i,\downarrow}\rangle] - t      \sum_{\langle i,j\rangle,\sigma} (c^{\dagger}_{i,\sigma}c_{j,\sigma}+h.c). $$

With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion



D(E_{\rm F})U > 1. $$