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. [1] one can rewrite (1) in the following form:

H = JmnSmSn +SmKm + SnKn + Eother,


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.

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

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