
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 electron-electron repulsion. 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,
where the second term accounts for the exchange energy, formula_2 is the Stoner parameter, formula_3 (formula_4) is the dimensionless density of spin up (down) electrons and formula_5 is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If formula_6 is fixed, formula_7 can be used to calculate the total energy of the system as a function of its polarization formula_8. If the lowest total energy is found for formula_9, the system prefers to remain paramagnetic but for larger values of formula_2, polarized ground states occur. It can be shown that for
the formula_9 state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the formula_9 density of states at the Fermi energy formula_14.
A non-zero formula_15 state may be favoured over formula_9 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 formula_17 plus fluctuation formula_18 and the product of spin-up and spin-down fluctuations is neglected. We obtain
With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion