Stoner criterion

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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[edit]

Ferromagnetism ultimately stems from electron-electron interactions. 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, N_\uparrow/N (N_\downarrow/N) is the dimensionless density[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_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[1] at the Fermi level D(E_F).

Note that a non-zero P state may be favoured over P=0 even before the Stoner criterion is fulfilled.

Relationship to the Hubbard model[edit]

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[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 +
       \sum_{i,\sigma} \epsilon_i n_{i,\sigma}.

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

  D(E_F)U > 1.



  • 1. Having a lattice model in mind, N is the number of lattice sites and 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, \epsilon(k) is replaced by discrete levels \epsilon_i and then D(E)=\sum_i \delta(E-\epsilon_i).