# 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,

$E_{\uparrow }(k)=\epsilon (k)-I{\frac {N_{\uparrow }-N_{\downarrow }}{N}},\qquad E_{\downarrow }(k)=\epsilon (k)+I{\frac {N_{\uparrow }-N_{\downarrow }}{N}},$ where the second term accounts for the exchange energy, $I$ is the Stoner parameter, $N_{\uparrow }/N$ ($N_{\downarrow }/N$ ) is the dimensionless density[note 1] 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[note 1] 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[note 1]

$H=U\sum _{i}n_{i,\uparrow }\langle n_{i,\downarrow }\rangle +n_{i,\downarrow }\langle n_{i,\uparrow }\rangle -\langle n_{i,\uparrow }\rangle \langle n_{i,\downarrow }\rangle -t\sum _{\langle i,j\rangle ,\sigma }(c_{i,\sigma }^{\dagger }c_{j,\sigma }+h.c).$ With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion

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