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.

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 (I is the Stoner parameter) , $N_{\uparrow }/N$ $N_{\uparrow }/N$ ( $N_{\downarrow }/N$ $N_{\downarrow }/N$ ) is the dimensionless density[1] of spin up (down) electrons and $\epsilon (k)$ $\epsilon (k)$ is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded. If $N_{\uparrow }+N_{\downarrow }$ $N_{\uparrow }+N_{\downarrow }$ is fixed, $E_{\uparrow }(k),E_{\downarrow }(k)$ $E_{\uparrow }(k),E_{\downarrow }(k)$ can be used to calculate the total energy of the system as a function of its polarization $P=(N_{\uparrow }-N_{\downarrow })/N$ $P=(N_{\uparrow }-N_{\downarrow })/N$ . 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

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