The Heisenberg model used in the JaSS is written in the form

H = Σi>jJijSiSj.                                                  (1)

If exchange constants are to be compared with models, where the summation runs twice over each pair of indexes (i.e. Σi>jJ'ijSiSj ), then 2J'ij=Jij.

In order to obtain exchange interaction parameters one can focus on single coupling Jmn between spins on m and n sites. Following Ref.  one can rewrite (1) in the following form:

H = JmnSmSn +SmKm + SnKn + Eother,

where

Km = Σi≠m,nJmiSi,

Kn = Σi≠m,nJniSi,

Eother = Σi,j≠m,nJijSiSj.

It should be noted that Km, Kn and Eother do not depend on the spin directions on sites m and n. Let’s consider four collinear spin configurations (with the spins along z quantization axis), where ↑m or ↓m denote spin direction on site m:

1) . . . ↑m . . . ↑n
2) . . . ↑m . . . ↓n
3) . . . ↓m . . . ↑n
4) . . . ↓m . . . ↓n

In these four spin states, the spin orientations on sites other than m and n are the same. These four states have the following energy:

E1 = Eother + JmnS2 + KmS + KnS,
E2 = Eother JmnS2+ KmS KnS,
E3 = Eother JmnS2KmS + KnS,
E4 = Eother + JmnS2KmS KnS.

Then one can obtain the expression for Jmn as

Jmn = (E 1 − E 2 − E 3 + E 4)/(4S2).

Computationally, this approach is rather inefficient since the calculation of N exchange interactions requires the energies of 4N spin configurations. But in complex magnets it turns out to be very useful for analysis of multiple, heavily intertwined exchanges.

 H. Xiang, E. Kan, S.-H. Wei, M.-H. Whangbo, X. Gong, Phys. Rev. B 84, 224429 (2011).