I am trying to understand physical intuition behind the Ising and Heisenberg models (thus I am not sure if my question is appropriate for this mostly mathematical site). I will concentrate of the Heisenberg model.

In the Heisenberg model of ferromagnetism the atoms are assumed to be arranged in a lattice. To the $i$-th atom corresponds a spin operator $\vec S_i$ (here $i$ belongs to the lattice). The Hamiltonian is given by $H=-J\sum_{<ij>}\vec S_i\cdot\vec S_j$ where the sum runs over all pairs of atoms connected by an edge, $J>0$ is a constant.

As far as I understand from the literature that this form of the Hamiltonian comes from the exchange interaction which, in turn, comes from the fact that each (neutral) atom is a boson (see e.g. vol. IX of Landau-Lifshitz, paragraph 72).

QUESTION. In addition to exchange interaction there exists a different spin-spin interaction which has a different form. Why it was neglected?

The spin-spin interaction between two atoms with magnetic moments $\vec\mu_1,\vec\mu_2$ and large separation $\vec R$ is equal to $$\frac{\vec\mu_1\cdot\vec\mu_2}{R^3}-3\frac{(\vec\mu_1\cdot\vec R)(\vec\mu_2\cdot\vec R)}{R^5}.$$ After quantization, $\vec \mu_i$ is replaced by $\vec S_i$ times a constant.

As it is claimed in "Statistical mechanics" by R. Feynman, Section 7.1, the spin-spin interaction is very small in comparison to the exchange interaction. However this seems to be true only on short distances since the exchange interaction decays exponentially with the distance $R$ between the atoms, while the spin-spin decays as $1/R^3$. Hence when we are on 3d-lattice the sum over the lattice of terms of order $1/R^3$ diverges logarithmically. This cannot be neglected from the experimental point of view too. When the ferromagnetic material is magnetized, it creates a magnetic field, which will interact with another magnet or moving charge. This interaction has already nothing to do with the exchange interaction, as far as I understand.


This doesn't seem quite like an appropriate question for a math site, so I guess it will be transferred shortly, but anyway, the answer to your question is that the dipolar interaction is by no means neglected in theories of ferromagnetism, it is the long-range interaction that governs the appearance of domain walls. The short-range exchange interaction, on the other hand, governs the appearance of ferromagnetic order within a domain wall.

Experimentally, it is clear that the dipolar interaction cannot be the origin of ferromagnetic order, because the corresponding critical temperature would be below 1 Kelvin, whereas typically the ferromagnetic order sets in far above room temperature. (For iron the critical temperature is 1043 Kelvin.) So you need the Heisenberg exchange to have ferromagnetic order to begin with, and then you can include dipolar interactions to see how the magnetic domains will form.

  • $\begingroup$ Many thanks! It would be very helpful to have a reference where this approach is discussed in greater detail. $\endgroup$
    – MKO
    Jan 28 '14 at 17:40

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