# 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,

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

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[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

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

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[1] at the Fermi level ${\displaystyle D(E_{F})}$.

Note that a non-zero ${\displaystyle P}$ state may be favoured over ${\displaystyle P=0}$ 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 ${\displaystyle \langle n_{i}\rangle }$ plus fluctuation ${\displaystyle n_{i}-\langle n_{i}\rangle }$ and the product of spin-up and spin-down fluctuations is neglected. We obtain[1]

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

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

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

## Footnotes

• 1. Having a lattice model in mind, ${\displaystyle N}$ is the number of lattice sites and ${\displaystyle N_{\uparrow }}$ is the number of spin-up electrons in the whole system. The density of states has the units of inverse energy. On a finite lattice, ${\displaystyle \epsilon (k)}$ is replaced by discrete levels ${\displaystyle \epsilon _{i}}$ and then ${\displaystyle D(E)=\sum _{i}\delta (E-\epsilon _{i})}$.