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,

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

where the second term accounts for the exchange energy, ${\displaystyle I}$  is the Stoner parameter, ${\displaystyle N_{\uparrow }/N}$  (${\displaystyle N_{\downarrow }/N}$ ) is the dimensionless density^{[note 1]} of spin up (down) electrons and ${\displaystyle \epsilon (k)}$  is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If ${\displaystyle N_{\uparrow }+N_{\downarrow }}$  is fixed, ${\displaystyle E_{\uparrow }(k),E_{\downarrow }(k)}$  can be used to calculate the total energy of the system as a function of its polarization ${\displaystyle P=(N_{\uparrow }-N_{\downarrow })/N}$ . If the lowest total energy is found for ${\displaystyle P=0}$ , the system prefers to remain paramagnetic but for larger values of ${\displaystyle I}$ , polarized ground states occur. It can be shown that for

${\displaystyle ID(E_{\rm {F}})>1}$ 

the ${\displaystyle P=0}$  state will spontaneously pass into a polarized one. This is the Stoner criterion, expressed in terms of the ${\displaystyle P=0}$  density of states^{[note 1]} at the Fermi energy ${\displaystyle D(E_{\rm {F}})}$ .

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

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 ${\displaystyle \langle n_{i}\rangle }$  plus fluctuation ${\displaystyle n_{i}-\langle n_{i}\rangle }$  and the product of spin-up and spin-down fluctuations is neglected. We obtain^{[note 1]}

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

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

${\displaystyle D(E_{\rm {F}})U>1.}$ 

• Stephen Blundell, Magnetism in Condensed Matter (Oxford Master Series in Physics).
• Teodorescu, C. M.; Lungu, G. A. (November 2008). "Band ferromagnetism in systems of variable dimensionality". Journal of Optoelectronics and Advanced Materials. 10 (11): 3058–3068. Retrieved 24 May 2014.
• Stoner, Edmund Clifton (April 1938). "Collective electron ferromagnetism". Proc. R. Soc. Lond. A. 165 (922): 372–414. Bibcode:1938RSPSA.165..372S. doi:10.1098/rspa.1938.0066.